Genericity conjecture for dope matrices

Let $P\in\mathbb{C}[x]$ be a polynomial, let $\Lambda=(\lambda_1,\ldots,\lambda_m)$ be an $m$-tuple of distinct complex numbers, and let $D_P(\Lambda)$ denote the associated $m\times(n+1)$ dope matrix. For generic $\Lambda$, write $\mathcal{D}_n^{\operatorname{gen}(m)}$ for the set of such matrices.

Genericity conjecture. For $m\geq 3$, $\mathcal{D}_n^{\operatorname{gen}(m)}$ consists exactly of the $m\times(n+1)$ $\{0,1\}$-matrices such that, for every $k\in[0,n]$, there are at most $k$ nonzero entries in the last $k+1$ columns.

The condition is known to be sufficient when $m=1$ and, by the paper's theorem, when $m=2$. Computations for $m=3$ and $n\leq 6$ support the conjecture, but the general case $m\geq 3$ remains open.