The generalized polynomial Wronskian conjecture

Let $P_1,\dots,P_m$ be $m$ polynomials of arbitrary degrees $0\leq d_1,\dots,d_m\leq m-1$ in $(\mathbb{F}[\alpha_1,\dots,\alpha_n])[\alpha]$. Define

$$W(P_1,\dots,P_m)\triangleq\det((C_{j,i})_{i,j\in[m]}),$$

where $P_i(\alpha)=\sum_{j\in[m]}C_{i,j}\alpha^{j-1}$, and say that a set has the generalized rectangular property (GRP) if, for some $1<k\leq m$ and $0\leq l\leq m-k$, at least $k$ polynomials of degrees at most $m-l-1$ have at least $m-k-l+1$ common roots. Generalized polynomial Wronskian conjecture. If $W(P_1,\dots,P_m)\equiv0$, then $\{P_1,\dots,P_m\}$ has GRP. This generalizes the equal-degree formulation and is equivalent to a generalized TM-MDS conjecture; its resolution status was not given in the source.