Lexicographic initial ideals conjecture for row-linear determinantal ideals

Let $L=(L_{ij})$ be an $m\times n$ matrix whose entries have the form

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

with $a_{ijk}\in K$ for all $i,j,k$. Assume that, for each $i$, the forms $L_{i1},\dots,L_{in}$ are linearly independent. Lexicographic initial ideals conjecture. Every lexicographic initial ideal of the ideal $I_2(L)$ of the $2$-minors of $L$ is generated in multidegree at most $(1,1,\dots,1)\in\mathbb Z^m$. This is the weaker form of the preceding conjecture that the paper identifies as sufficient for its Koszulness argument.