Support conjecture for mirror duals of Procesi bundles

Let $S$ be the surface in the Hilbert-scheme setup, let $\mathcal{M}_n$ be the corresponding moduli space, and let $I\in\mathcal{M}_n^{\mathbb{T}}$ be a fixed ideal. Let $\mathcal{P}_I$ be the associated Procesque bundle, $\mathcal{W}_I$ its mirror dual, and $W_I^{\ggcurly}$ the Lagrangian closure of $I$. Support conjecture. The mirror dual $\mathcal{W}_I$ is supported on

$$W_I^{\ggcurly}.$$

This extends the expected support statement beyond very stable components; computing the equivariant multiplicities of $\mathcal{W}_I$ along the irreducible components of this closure remains open.