Promotion conjecture for tensor products of Kirillov–Reshetikhin crystals

Let $B=B^{r_k,s_k}\otimes\cdots\otimes B^{r_1,s_1}$ be a tensor product of Kirillov–Reshetikhin crystals, and let $L$ be its multiplicity array. Let $\mathrm{RC}(L)$ be the corresponding set of unrestricted rigged configurations, and let $\mathrm{pr}:\mathrm{RC}(L)\to\mathrm{RC}(L)$ be the promotion operator defined by the stated crystal operators and the algorithm $\rho$. For a single factor $B^{r,s}$, the theorem in the source asserts that this map is the promotion operator on rigged configurations.

Promotion conjecture. Theorem 4.11 remains true for any tensor product

$$B=B^{r_k,s_k}\otimes\cdots\otimes B^{r_1,s_1}.$$

Thus the map defined by the promotion algorithm should be the promotion operator on $\mathrm{RC}(L)$ for every such tensor product. The source presents this as Conjecture 4.12 and supplies no resolution evidence.