The limiting Van Vleck measure conjecture for generalized Lamé equations

Let $Q(z)$ and $P(z)$ be polynomials with $\deg Q(z)=l\geq 2$ and $\deg P(z)\leq l-1$, and let $\mathcal V_n$ be the finite measure on the space $Pol_{l-2}$ of monic polynomials of degree $l-2$ formed by the normalized Van Vleck polynomials whose associated Stieltjes polynomial has degree exactly $n$, with multiplicities divided by $\binom{n+l-2}{n}$. Van Vleck measure conjecture. The sequence $\{\mathcal V_n\}$ of finite measures converges to a probability measure $\mathcal V_Q$ in $Pol_{l-2}$ which depends only on the leading coefficient $Q(z)$. This is presented as a weaker version of a conjecture of the second author; the supplied context indicates that the limiting-measure question is related to the still completely open problem of developing Strebel differentials for quadratic differentials of order greater than two.