Spectral-measure convergence conjecture for higher Lamé operators

Let $\mathfrak d(z)=\sum_{i=1}^k Q_i(z)\frac{d^i}{dz^i}$ be a non-degenerate higher Lamé operator, and let $r=\max_{i=1,\ldots,k}(\deg Q_i-i)$. Let $Pol_r$ be the space of monic polynomials of degree $r$. For each sufficiently large $n$, let $\sigma_n(\mathfrak d(z))$ be the finite measure on $Pol_r$ assigning equal mass to the normalized Van Vleck polynomials associated with degree-$n$ Stieltjes polynomials. Spectral-measure convergence conjecture. For any non-degenerate $\mathfrak d(z)$, the sequence $\sigma_n(\mathfrak d(z))$ converges weakly to a compactly supported measure $\Sigma(\mathfrak d(z))$ on $Pol_r$, and $\Sigma(\mathfrak d(z))$ depends only on the leading monomial $Q_k(z)\frac{d^k}{dz^k}$. This conjecture asserts existence and leading-term universality of the limiting Van Vleck measure; the supplied text gives no resolution.