of places in $L$ lying over a place in $\tilde{\Sigma}_p^+$, $\tilde{\Sigma}_{L,0}$ the set of places in $L$ lying over a place in $\tilde{\Sigma}_0^+$, and $\tilde{\Sigma}_{L,1}$ be the set of places in $L$ lying over $\tilde{v}_1$. Let $\tilde{\Sigma}_L$ be the union of $\tilde{\Sigma}_{L,p}$ and $\tilde{\Sigma}_{L,0}$. If $\tilde{v} \in \tilde{\Sigma}_{L,p}$, let $\tau_{\tilde{v}} = \tau_{\tilde{v}}|_{F^+}|_{I_L}$. If $\tilde{v} \in \tilde{\Sigma}_{L,0}$, let $R^{\square, \tau_{\tilde{v}}}$ be the lifting ring for $\tilde{r}|_{G_{L,\tilde{v}}}^-$ parametrizing lifts whose characteristic polynomial is $(X-1)^3$. We let $R_{\tau}^{\text{univ}}$ be the universal deformation ring corresponding to the deformation problem

$$ \left( F/F^+, S, \tilde{S}, \mathcal{O}_E, \bar{r}, \epsilon^{-2}\delta_{F/F^+}, \{R_{\tilde{v}_1}^{\square}\} \cup \{R_{\tilde{v}}^{\square, \tau_{\tilde{v}}}\}_{v \in \Sigma^+} \right), $$

and we let $R_S^{\text{univ}}$ be the universal deformation ring corresponding to the deformation problem

$$ \left( L/L^+, S_L, \tilde{S}_L, \mathcal{O}_E, \bar{r}|_{G_L}, \epsilon^{-2}\delta_{L/L^+}, \{R_{\tilde{u}_1}^{\square}\}_{\tilde{u}_1 \in \tilde{S}_{L,1}} \cup \{R_{\tilde{v}}^{\square, \tau_{\tilde{v}}}\}_{\tilde{v} \in \tilde{S}_L} \right). $$

The proof of [Gee11, Theorem 5.1.4] shows that $R_{\tau}^{\text{univ}}$ is finite over $\mathcal{O}_E$. Indeed, $R_{\tau}^{\text{univ}}$ is finite over $R_S^{\text{univ}}$ and $R_S^{\text{univ}}$ is finite over $\mathcal{O}_E$ since $(R_S^{\text{univ}})^{\text{red}}$ is isomorphic to an appropriate Hecke algebra by the proof of [Gue11, Theorem 3.4]. One replaces Fontaine–Laffaille deformation rings with $R^{\square, \tau_{\tilde{v}}}$ for $\tilde{v} \in \tilde{\Sigma}_{L,p}$, which is geometrically integral by [LLHLM18, Section 5.3].

That $R_{\tau}^{\text{univ}}$ is finite over $\mathcal{O}_E$ implies that there is a conjugate self-dual lift $r: G_F \to GL_3(\overline{\mathbb{Q}}_p)$ of $\bar{r}$ which is minimally ramified outside $p$ and potentially crystalline of type $((0, 1, 2), \tau_v)$ at $\tilde{v}$ for each $v \in \Sigma_0^+$. Moreover, we have that the restriction $r|_{G_L}$, which corresponds to a point of Spec $R_S^{\text{univ}}$, is automorphic. Solvable base change then implies that $r$ is automorphic. Local–global compatibility implies that $S(U, \sigma \otimes W)_m$ is nonzero for $U$ and $W$ as in the statement of the proposition. Then $S(U, V' \otimes W)_m$ is nonzero for some reachable $V' \in \text{JH}(\bar{\sigma})$. $\square$

This would then show that $M_{\infty}(\otimes_{v \in \Sigma_p^+} \bar{\sigma}(\tau_v)^{\circ})$ is nonzero in the third paragraph (one cannot directly cite Definition 7.11(2) because of the change above).

(17) In Section 8.1, after the proof of Lemma 8.2, the $\mathcal{O}$-algebra $R_{\overline{\mathfrak{M}, \bar{\rho}}}^{\tau, \bar{\beta}, \square}$ has relative dimension 15 over $\mathcal{O}$.

(18) In Corollary 8.4, there should be the further relation $c_{12}c_{33} = 0$.

(19) The final sentence of Corollary 8.4 should be replaced by the claim that the ring $R_{\overline{\mathfrak{M}, \bar{\rho}}}^{\tau, \bar{\beta}, \square}/\sigma$ is a quotient of $\tilde{R}[\llbracket c_{ii}^* - [\tilde{c}_{ii}^*], x_j \rrbracket, 1 \le i \le 3, 1 \le j \le 3]$.
9]]. This comment applies also to Proposition 8.11. These changes justify the dimension hypothesis in Lemma 8.8, which is used in the proof of Proposition 8.6.

(20) In the proof of Proposition 8.11, we remark that $c_{33} = c_{23}c_{31}(c_{21}^*)^{-1}$.

(21) In the caption of Table 4, the coefficients are in $\mathbb{F}$.

(22) In the caption of Table 5, the coefficients are in $\mathbb{R}$.

(23) The entry (1, 3) of Table 6 should read 'Leading term of the monodromy condition'.

(24) The missing entries in the second column of Table 6 can be read off from Table 5 and the missing entry in the third column of Table 6 can be read off from Proposition 8.3.

(25) In Table 6, the leading term of the monodromy condition for $\alpha\beta\alpha\gamma$, the second $c_{33}^*$ should be removed. Moreover, in the caption, the $a_{s_{j+1}(i)}^{(j)}$ for $i=1, 2, 3$ should be bold.

## Acknowledgements

We would like to thank Christophe Breuil, James Humphreys, and Cornelius Pillen for many helpful conversations. We would also like to thank Matthew Emerton, Toby Gee, and Florian Herzig for their support, guidance, and for comments on an earlier draft of this paper. Part of this work was carried out while the authors were visiting the Institut Henri Poincaré and the Mathematisches Forschungsinstitut Oberwolfach, and we would like to thank these institutions for their hospitality. BLH acknowledges support from the National Science Foundation under grant Nos. DMS-1128155, DMS-1802037 and the Alfred P. Sloan Foundation. DL was supported by the National Science Foundation under agreement Nos. DMS-1128155 and DMS-1703182 and an AMS-Simons travel grant. Finally, the authors express their utmost gratitude to the anonymous referee for his or her meticulous and patient reading of several versions of this paper. The reports were invaluable in helping us to improve the quality, precision, and clarity of this paper.

**Conflict of Interest:** The authors declare no conflict of interest.
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