Stability conjecture for the combinatorial R-matrix on the limiting crystal

Let $B_\natural$ be the limiting crystal obtained from the crystals $B_l$, and let $g_l:B_l\to B_\natural$ be its set-theoretic embedding. For fixed $u\in B_\natural$ and $b\in B_1$, define $c(l,u,b)\in B_1$ and $v(l,u,b)\in B_l$ by

$$u_l\otimes b\simeq c(l,u,b)\otimes v(l,u,b),$$

where $u_l\in B_l$ is the unique element satisfying $g_l(u_l)=u$ for sufficiently large $l$, and the isomorphism is the combinatorial $R$-matrix $B_l\otimes B_1\simeq B_1\otimes B_l$. Stability conjecture. For any fixed $u\in B_\natural$ and $b\in B_1$, the elements $c(l,u,b)\in B_1$ and $g_l\bigl(v(l,u,b)\bigr)\in B_\natural$ are independent of $l$ for all sufficiently large $l$. This asserts that the combinatorial $R$-matrix has a well-defined stable limit on the limiting crystal; the source gives no proof or resolution in the supplied text.