  The Willmore Problem seeks the surface in $\mathbb{S}^3\subset\mathbb{R}^4$
of a given topological type minimizing the squared-mean-curvature energy $W =
\int |\mathbf{H}_{\mathbb{R}^4}|^2 = \operatorname{area} + \int
H_{\mathbb{S}^3}^2$. The longstanding Willmore Conjecture that the Clifford
torus minimizes $W$ among genus-$1$ surfaces is now a theorem of Marques and
Neves [19], but the general conjecture [10] that Lawson's [16] minimal surface
$\xi_{g,1}\subset\mathbb{S}^3$ minimizes $W$ among surfaces of genus $g>1$
remains open. Here we prove this conjecture under the additional assumption
that the competitor surfaces $M\subset\mathbb{S}^3$ share the ambient
symmetries of $\xi_{g,1}$. More generally, we show each Lawson surface
$\xi_{m,k}$ satisfies the analogous $W$-minimizing property under a somewhat
smaller symmetry group ${G}_{m,k}<SO(4)$, using a local computation of the
orbifold Euler number $\chi_o(M/{G}_{m,k})$ to exclude certain intersection
patterns of $M$ with the great circles fixed by generators of ${G}_{m,k}$.
