Characterization of non-varying strata of quadratic differentials

Let $\mathcal{Q}_g(d_1,\ldots,d_n)$ be a stratum of quadratic differentials, and let $L^+$ denote its value on the stratum. Assume that $\mathcal{Q}_g(d_1,\ldots,d_n)$ is neither hyperelliptic nor irregular. Define

$$S=\left\{\frac{2k}{d_j+2}\mathrel{|}0<2k\leq d_j+1,\\ j=1,\ldots,n\right\}.$$

Non-varying-stratum characterization. The stratum $\mathcal{Q}_g(d_1,\ldots,d_n)$ is non-varying if and only if $L^+$ equals the sum of the $g$ smallest numbers in $S$. This is presented as a speculation based on the authors’ results and evidence; determining whether any new non-varying strata exist beyond those previously known remains open.