Sheaf quantization conjecture for the Givental–Hori–Vafa mirror

Let $X_\Sigma$ be the toric variety with Givental–Hori–Vafa mirror $W_\Sigma$, and let $\widehat{\Lambda}_{\geq}$ denote the Novikov ring. Write $\mathrm{SQ}(W_\Sigma,\widehat{\Lambda}_{\geq})$ for the subcategory spanned by sheaf quantizations of end-conic Lagrangians ending in $\widehat{\Lambda}_{\geq}$. Sheaf quantization conjecture. The category

$$\mathrm{SQ}(W_\Sigma,\widehat{\Lambda}_{\geq})$$

is equivalent to the derived Fukaya category of $T^*(M_{\mathbb R}/M)$ over the Novikov ring generated by Lagrangian submanifolds compactly Hamiltonian isotopic to Lefschetz thimbles of $W_\Sigma$. This proposes a genuine sheaf-theoretic analogue of the Fukaya category over the Novikov ring, but the source gives no evidence that the equivalence is known.