Take a linear ordinary differential operator $\mathfrak{d}\left( z \right) = \sum\nolimits_{i = 1}^k {Q_i \left( z \right)\frac{{d^i }} {{dz^i }}}$ with polynomial coefficients and set r = max i=1,…,k(deg Q i (z) − i). If d(z) satisfies the conditions: (i) r ≥ 0 and (ii) deg Q k (z) = k + r, we call it a non-degenerate higher Lamé operator. Following the classical examples of E. Heine and T. Stieltjes we initiated in [13] the study of the following multiparameter spectral problem: for each positive integer n find polynomials V (z) of degree at most r such that the equation $\mathfrak{d}\left( z \right)S\left( z \right) + V\left( z \right)S\left( z \right) = 0$ has a polynomial solution S(z) of degree n. We have shown that under some mild non-degeneracy assumptions on T there exist exactly $\left( {\begin{array}{*{20}c} {n + r} \\ n \\ \end{array} } \right)$ spectral polynomials V n,i (z) of degree r and their corresponding eigenpolynomials S n,i (z) of degree n. Localization results of [13] provide the existence of abundance of converging as n→∞ sequences of normalized spectral polynomials $\left\{ {\tilde V_{n,i_n } \left( z \right)} \right\}$ where $\tilde V_{n,i_n } \left( z \right)$ is the monic polynomial proportional to $V_{n,i_n } \left( z \right)$ . Below we calculate for any such converging sequence $\left\{ {\tilde V_{n,i_n } \left( z \right)} \right\}$ the asymptotic root-counting measure of the corresponding family $\left\{ {S_{n,i_n } \left( z \right)} \right\}$ of eigenpolynomials. We also conjecture that the sequence of sets of all normalized spectral polynomials $\left\{ {\tilde V_{n,i} \left( z \right)} \right\}$ having eigenpolynomials S(z) of degree n converges as n→∞ to the standard measure in the space of monic polynomials of degree r which depends only on the leading coefficient Q k (z).