The cohomological conjecture for Dirac matrix factorizations

Let $\triangle$ be a combinatorially spin Delzant polytope such that the real toric locus $X_\triangle^\mathbb{R}$ is Fano. Let $M_\triangle$ be the associated Dirac matrix factorization, let $\operatorname{End}(M_\triangle)_0$ denote its degree-zero endomorphism complex, and let $(R_\triangle)_0$ be the degree-zero part of the coefficient ring. Let $H^*$ denote cohomology.

Cohomological conjecture. For each combinatorially spin Delzant polytope $\triangle$ such that $X_\triangle^\mathbb{R}$ is Fano, we have

$$H^*(\operatorname{End}(M_\triangle)_0)\cong H^*(X_\triangle^\mathbb{R}; (R_\triangle)_0).$$

This is a cohomological mirror-symmetry prediction relating the Dirac matrix factorization to the real toric locus. The supplied text states that the claim is proved for $\triangle=\triangle^n$ when $n$ is odd, while the general case remains unresolved.