Crystal conjecture on radicality of join-meet ideals

Let $L_k(n_1,n_2,\ldots,n_k)$ be the lattice considered in the paper, and let $I_{L_k(n_1,n_2,\ldots,n_k)}$ denote its join-meet ideal in the associated polynomial ring. Crystal conjecture. The join-meet ideal

$$I_{L_k(n_1,n_2,\ldots,n_k)}$$

is radical. This conjecture is motivated by the radical decompositions established for the explicitly computed families in Theorems 1 and 2; its status in the generality stated here is not specified in the source.