The characterization of MDS-PIR capacity-achieving codes

Let $\mathcal{C}$ be an $[n,k]$ code, and let $d_s^{\mathcal{C}}$ denote the quantity appearing in the necessary condition

$$d_s^{\mathcal{C}}\geq\frac{n}{k}s,\quad\forall\,s\in\{1,\ldots,k\}.$$

An MDS-PIR capacity-achieving matrix $\bm{\Lambda}_{\kappa,\nu}(\mathcal{C})$ has parameters satisfying $\frac{\kappa}{\nu}=\frac{k}{n}$. MDS-PIR characterization conjecture. Such a matrix exists if and only if

$$d_s^{\mathcal{C}}\geq\frac{n}{k}s,\quad\forall\,s\in\{1,\ldots,k\}.$$

The condition is necessary for the existence of an MDS-PIR capacity-achieving matrix. Exhaustive searches for parameters up to length $11$, with the stated exceptions $[n,k]=[10,5]$ and $[n,k]=[11,4\leq k\leq 7]$, found a matrix whenever the condition held, but sufficiency remains open in general.