The case $d > 0$. Let $K^{-1} \to K^0$ be a complex representing $K$ as in More on Algebra, Lemma \ref{more-algebra-lemma-ext-1-annihilated-definite} part (5) with respect to the ideal $I^c/I^n$ in the ring $A/I^n$. In particular $K^{-1}$ is $I^c/I^n$-projective as multiplication by elements of $I^c/I^n$ even factor through $K^0$. By More on Algebra, Lemma \ref{more-algebra-lemma-map-out-of-almost-free} part (1) we have $$ \Ext^1_{A/I^n}(K, I^n/I^{2n}) = \Coker(\Hom_{A/I^n}(K^0, I^n/I^{2n}) \to \Hom_{A/I^n}(K^{-1}, I^n/I^{2n})) $$ and similarly for other Ext groups. Hence any class $\xi$ in $\Ext^1_{A/I^n}(K, I^n/I^{2n})$ comes from an element $\varphi \in \Hom_{A/I^n}(K^{-1}, I^n/I^{2n})$. Denote $\varphi'$ the image of $\varphi$ in $\Hom_{A/I^n}(K^{-1}, I^{n - (1 + d)c}/I^{2n - (1 + d)c})$. By Lemma \ref{lemma-factor-hom} we can write $\varphi' = \sum a_i \psi_i$ with $a_i \in I^c$ and $\psi_i \in \Hom_{A/I^n}(M, I^{n - (2 + d)c}/I^{2n - (2 + d)c})$. Choose $h_i : K^0 \to K^{-1}$ such that $a_i \text{id}_{K^{-1}} = h_i \circ d_K^{-1}$. Set $\psi = \sum \psi_i \circ h_i : K^0 \to I^{n - (2 + d)c}/I^{2n - (2 + d)c}$. Then $\varphi' = \psi \circ \text{d}_K^{-1}$ and we conclude that $\xi$ already maps to zero in $\Ext^1_{A/I^n}(K, I^{n - (1 + d)c}/I^{2n - (1 + d)c})$ and a fortiori in $\Ext^1_{A/I^n}(K, I^{n - (1 + d)c}/I^{2n - 2(1 + d)c})$. \medskip\noindent The case $d = 0$\footnote{The argument given for $d > 0$ works but gives a slightly weaker result.}. Let $\xi$ and $\varphi$ be as above. We consider the diagram $$ \xymatrix{ K^0 \\ K^{-1} \ar[u] \ar[r]^\varphi & I^n/I^{2n} \ar[r] & I^{n - c}/I^{2n - c} } $$ Pulling back to $X$ and using the map $p^*(I^n/I^{2n}) \to \mathcal{O}_{Y_n}(n)$ we find a solid diagram $$ \xymatrix{ p^*K^0 \ar@{..>}[rrd] \\ p^*K^{-1} \ar[u] \ar[r] & \mathcal{O}_{Y_n}(n) \ar[r] & \mathcal{O}_{Y_n}(n - c) } $$ We can cover $X$ by affine opens $U = \Spec(B)$ such that there exists an $a \in I$ with the following property: $IB = aB$ and $a$ is a nonzerodivisor on $B$. Namely, we can cover $X$ by spectra of affine blowup algebras, see Divisors, Lemma \ref{divisors-lemma-blowing-up-affine}. The restriction of $\mathcal{O}_{Y_n}(n) \to \mathcal{O}_{Y_n}(n - c)$ to $U$ is isomorphic to the map of quasi-coherent $\mathcal{O}_U$-modules corresponding to the $B$-module map $a^c : B/a^nB \to B/a^nB$. Since $a^c : K^{-1} \to K^{-1}$ factors through $K^0$ we see that the dotted arrow exists over $U$. In other words, locally on $X$ we can find the dotted arrow! Now the sheaf of dotted arrows fitting into the diagram is principal homogeneous under $$ \mathcal{F} = \SheafHom_{\mathcal{O}_X}( \Coker(p^*K^{-1} \to p^*K^0), \mathcal{O}_{Y_n}(n - c)) $$ which is a coherent $\mathcal{O}_X$-module. Hence the obstruction for finding the dotted arrow is an element of $H^1(X, \mathcal{F})$. This cohomology group is zero as $1 > d = 0$, see discussion following the definition of $d = d(S)$. This proves that we can find a dotted arrow $\psi : p^*K^0 \to \mathcal{O}_{Y_n}(n - c)$ fitting into the diagram. Since $n - c \geq q_0$ we find that $\psi$ induces a map $K^0 \to I^{n - c}/I^{2n - c}$. Chasing the diagram we conclude that $\varphi' = \psi \circ \text{d}_K^{-1}$ and the proof is finished as before.