Nonsingularity conjecture for t-wise intersection matrices

For a positive integer $n$, write $[n]=\{1,\ldots,n\}$. For subsets $I_1,\ldots,I_t\subseteq[n]$, define their weight by

$$\operatorname{wt}(I_1,\ldots,I_t)=\sum_{i=1}^t|I_i|-\left|\bigcup_{i=1}^t I_i\right|,$$

and write $\operatorname{wt}(I_J)=\operatorname{wt}(I_j:j\in J)$ for $J\subseteq[t]$. Let $M_{k,(I_1,\ldots,I_t)}$ denote the associated $t$-wise intersection matrix. Nonsingularity conjecture for t-wise intersection matrices. If $t\geq3$, the subsets $I_1,\ldots,I_t\subseteq[n]$ satisfy

$$\operatorname{wt}(I_J)\leq (|J|-1)k$$

for every nonempty $J\subseteq[t]$, and

$$\operatorname{wt}(I_{[t]})=(t-1)k,$$

then $M_{k,(I_1,\ldots,I_t)}$ is nonsingular over every finite field. This conjecture is the key algebraic condition underlying the capacity conjecture for Reed–Solomon list decoding, but remains unresolved.