Polynomiality conjecture for Schur classes of tautological line bundles

Let $S$) be a smooth connected surface with $h^1(O_S)=0$ whose Picard group is generated by line bundles $\mathcal{L}_1,\cdots,\mathcal{L}_m$. Consider a line bundle $\mathcal{L}=\mathcal{L}_1^{a_1}\otimes\cdots\otimes\mathcal{L}_k^{a_m}$, and let $S^{[n]}$ denote the Hilbert scheme of $n$ points on $S$. A Schur class of $\mathcal{L}^{[n]}$ of codimension $k$ is a class obtained by applying a Schur polynomial to the Chern classes of the tautological bundle $\mathcal{L}^{[n]}$. Polynomiality conjecture. The coefficients of any such Schur class, in any basis for the Chow ring of $S^{[n]}$, are polynomials of degree at most $k$ in the integers $a_i$. The conjecture records a general polynomial pattern in the coefficients of Schur classes of tautological line bundles; the supplied text gives no resolution or further evidence beyond this observed pattern.