Rank identifications in join-meet ideals of thin distributive lattices

Let $L$ be a thin distributive lattice of rank $m+1$, and let $I_L$ be its join-meet ideal in the polynomial ring $S_L$. For $A\subseteq\{1,\dots,m\}$, let $J_A$ be the ideal obtained from $I_L$ by identifying, for every $i\in A$, the variables of the two elements of $L$ of rank $i$. We prove that $