Tilting-object conjecture for canonical bundles over reflexive polytopes

Let $P$ be a reflexive polytope. Let $\Sigma$ be a simplicial fan whose ray primitive generators are the vertices of $P$, and let $\mathcal{V}$ be the fan of the canonical bundle over $X_\Sigma$. Reflexive-polytope tilting conjecture. If the toric Deligne–Mumford stack $\mathcal{X}_\Sigma$ has a tilting object $\mathcal{T}$, then $\mathcal{X}_\mathcal{V}$ also has a tilting object. Consequently, if $\mathcal{X}_\Sigma$ has a tilting object, then

$$R=k[|\operatorname{Cone}(P\times\{1\})|^\vee\cap M]$$

has an NCCR. The conjecture is motivated by expected cohomology vanishing for toric Fanos; the source gives no resolution.