The admissibility decomposition conjecture for tensor-product crystals

Let $B$ be the crystal under consideration, let $I$ be the index set of its simple coroots, and let $(P_{cl}^+)_l$ denote the dominant classical weights of level $l$. For $i\in I$, write $\varepsilon_i$ and $\varphi_i$ for the crystal string functions, and let $\widetilde{f}_i$ be the Kashiwara lowering operator. The pair $(\xi,b)$ is called admissible when it satisfies the admissibility condition used to define the restricted 1dsums.

Admissibility decomposition conjecture. For any $\xi\in(P_{cl}^+)_l$, there exists a disjoint union decomposition, not necessarily unique,

$$\{b\in B\mid(\xi,b)\text{ is not admissible}\} =\bigsqcup_{\substack{b'\in B,\varepsilon_i(b')=\langle h_i,\xi\rangle+1\text{ for some }i\in I}}\{\widetilde{f}_i^{\,t}b'\mid 0\leq t\leq\varphi_i(b')\}.$$

This decomposition is intended to describe precisely the non-admissible elements as crystal strings beginning at elements violating one of the level bounds. It is the assumption needed to express the restricted 1dsums in terms of the functions $g_j$; the supplied text gives no resolution, so its status remains open.