Positivity conjecture for principal minors of the bivariate Bézier collocation matrix

Let $d$ be a nonnegative integer, let $T$ be a triangle, and let $\mathcal I_d$ be the set of weak $3$-compositions of $d$. For each $\bfm i\in\mathcal I_d$, let $\xi_{\bfm i}$ be the corresponding domain point in barycentric coordinates, and let $B^d_{\bfm j}$ be the degree-$d$ Bernstein basis polynomial. For a nonempty subset $\Gamma=\{\bfm i_1,\dots,\bfm i_n\}\subset\mathcal I_d$, define

$$M_\Gamma=[B^d_{\bfm j}(\xi_{\bfm i})]_{\bfm i,\bfm j\in\Gamma}.$$

The principal-minor positivity conjecture. For every such $\Gamma$, the matrix $M_\Gamma$ is nonsingular and

$$\det M_\Gamma>0.$$

The conjecture asserts unique solvability and positivity for all constrained Lagrange interpolation problems based on these Bernstein collocation matrices. The paper states that the full conjecture is confirmed in its results, including arbitrary $\Gamma\subset\mathcal I_d$ for $d\leq17$; the supplied status is unknown, so its resolution beyond those cases should be checked.