Polynomial growth conjecture for component ideal degrees of matrix rigidity varieties

Fix $0<\epsilon<1$ and $0<\delta<1$. Set $r=\epsilon n$ and $s=n^{1+\delta}$. The notation $\hat{\mathcal R}[n,r,s]$ denotes the relevant matrix rigidity variety, and its irreducible components have ideals whose minimal polynomial degrees are being considered. Polynomial growth conjecture. The minimal degree of a polynomial in the ideal of each irreducible component of $\hat{\mathcal R}[n,r,s]$ grows like a polynomial in $n$. This predicts that, despite the increasing complexity of the associated cones, the defining equations of every irreducible component admit polynomially bounded degree. The source gives no resolution of this conjecture.