Square-free initial ideals conjecture for generic row-linear matrices

Let $t_{ij}$ be distinct variables over a field $K$ with $1\leq i\leq m$ and $1\leq j\leq n$. Let $L=(L_{ij})$ be an $m\times n$ matrix with

$$L_{ij}=\sum_{k=1}^n a_{ijk}t_{ik},$$

where $a_{ijk}\in K$ for all $i,j,k$. Denote by $I_2(L)$ the ideal of the $2$-minors of $L$. Square-free initial ideals conjecture. For every choice of the coefficients $a_{ijk}$ and every term order $<$ on $K[t_{ij}]$, the initial ideal

$$\operatorname{in}_<(I_2(L))$$

is square-free in the $\mathbb Z^m$-graded sense: it is generated by elements of the form $t_{i_1j_1}\cdots t_{i_kj_k}$ with $i_1<i_2<\cdots<i_k$. This would provide the replacement for the Sturmfels–Villarreal theorem needed to extend the Koszulness argument from the monomial case to general collections of linear spaces.