Cox category tensor-product conjecture for semiprojective toric varieties

Let $\Xfr$ be a smooth semiprojective toric variety. The category $\Sh_{\Lambda_\mathrm{Cox}}(M_\RR / M)$ is closed under $\otimes_k$. Cox category tensor-product conjecture. There is a symmetric monoidal equivalence

$$\big(\QC_{\mathrm{Cox}}(\Xfr), \star'\big) \simeq \big(\Sh_{\Lambda_\mathrm{Cox}}(M_\RR / M), \otimes_k\big)$$

where $\star'$ is a “birationally glued EC product” extending the convolution on the maximal torus of $\Xfr$. This is proposed as a categorical and homological-mirror-symmetry description of the Cox category, whose geometric interpretation involves gluing birational models of $\Xfr$; the source does not establish the conjecture or provide evidence resolving it.