Squarefree initial ideal conjecture for stretched lattice join-meet ideals

Let $k\geq 3$ and let $(n_1,\ldots,n_k)\neq(1,\ldots,1)$. Let $L_k(n_1,\ldots,n_k)$ be the stretched finite lattice with chains

$$s<a_{i,1}<a_{i,2}<\cdots<a_{i,n_i}<t$$

for $i=1,\ldots,k$, and let $I_{L_k(n_1,\ldots,n_k)}$ be its join-meet ideal. Squarefree initial ideal conjecture. There exists a monomial order $\prec'$ such that $I_{L_k(n_1,\ldots,n_k)}$ is squarefree with respect to $\prec'$. The proposed monomial-order formulation is motivated by the proof for $k=2$ and is presented as a natural conjecture for $k\geq 3$.