The Ext-enhanced monoidal structure conjecture for free-monodromic tilting sheaves

Let $\mathsf{FM}_2^{\mathrm{Ext}}$ be the Ext-enhanced category of free-monodromic complexes, and let $\mathrm{Tilt}_2^{\mathrm{Ext}}$ be the full subcategory consisting of $\widetilde{\mathcal{T}}_{\underline{w}}\langle n\rangle\llbracket m\rrbracket$ for expressions $\underline{w}$ and integers $n,m$. Write $\mathbin{\widehat{\star}}$ for the Ext-enhanced free-monodromic convolution and let $\widetilde{\mathcal{T}}_\varnothing$ denote the identity object. The previously defined category $\mathrm{Tilt}_2$ has the monoidal structure $(\mathbin{\widehat{\star}},\widetilde{\mathcal{T}}_\varnothing)$. Ext-enhanced monoidal structure conjecture. The triple $(\mathrm{Tilt}_2^{\mathrm{Ext}},\mathbin{\widehat{\star}},\widetilde{\mathcal{T}}_\varnothing)$ admits a monoidal structure extending that on $(\mathrm{Tilt}_2,\mathbin{\widehat{\star}},\widetilde{\mathcal{T}}_\varnothing)$. This conjecture proposes the additional grading enhancement of monoidal Koszul duality and, in particular, requires the Ext-enhanced convolution to satisfy the coherence and bifunctoriality needed for a monoidal category. The source gives no resolution of the conjecture.