In the survey https://www.aimath.org/WWN/kostka/crysdumb.pdf the following statement is stated as a Conjecture 4.5 (due to Kashiwara): "Every connected affine crystal graph is isomorphic to a tensor product of Kirillov-Reshetikhin crystals" (the conjecture is about finite affine type $A$ crystals).
The question is the following: is there any progress towards proving this conjecture? Maybe some weaker versions of it are proven?
For example is it known that if $B$ is a finite affine type $A_{n-1}^{(1)}$-crystal that is connected as $A_{n-1}$-crystal then $B$ is isomorphic to Kirillov-Reshetikhin crystal?
Or is it true that if $B$ is a finite affine (type $A_{n-1}^{(1)}$) connected crystal that as $A_{n-1}$-crystal is isomorphic to the tensor product of $A_{n-1}$-crystals corresponding to rectangular diagrams then the crystal $B$ is isomorphic to the tensor product of the corresponding Kirillov-Reshetikhin crystals? In the case when there is only one crystal in the tensor product (i.e. when $B$ is connnected as $A_{n-1}$-crystal) the claim follows for example from Lemma 2.6 of https://arxiv.org/pdf/1104.2359.pdf).
