Filling-map factorization conjecture for tensor products of KR crystals

Let

$$B=\bigotimes_{i=1}^N B^{r_i,s_i}$$

and let

$$T=\bigotimes_{i=1}^N (B^{1,1})^{\otimes r_i s_i}$$

be organized into $r_i\times s_i$ rectangles as in $T^{r_i,s_i}$. Let $\operatorname{fill}\colon B\to T$ be the filling map. Filling-map factorization conjecture. The filling map is the tensor product of the factorwise filling maps:

$$\operatorname{fill}(B)=\bigotimes_{i=1}^N\operatorname{fill}(B^{r_i,s_i}).$$

This extends the filling-map description from individual KR crystals to arbitrary tensor products; the general statement remains open.