The Day convolution compatibility conjecture for admissible double -categories

Let $\mathcal{X} \to \Delta^{\operatorname{op}}$ be an admissible double $\infty$-category and let $\mathcal{V}$ be a presentably monoidal $\infty$-category. Corollary 3.?? equips $\operatorname{Fun}(\mathcal{X}_{1}^{\operatorname{op}},\mathcal{V})$ with a monoidal structure. The presentable tensor product $\mathcal{P}(\mathcal{X}_{1}) \otimes \mathcal{V}$ also has a monoidal structure induced by the monoidal structure on $\mathcal{V}$ and the Day convolution on $\mathcal{P}(\mathcal{X}_{1})$. Day convolution compatibility conjecture. These two monoidal structures are equivalent. This predicts that the explicit monoidal structure constructed from the admissible double $\infty$-category agrees with the standard tensor-product and Day-convolution construction; the source gives no resolution of the expectation.