Fixed-order MLDE conjecture for generalized Schur partition functions

Let a 4d ${\cal N}=2$ superconformal field theory have rank $r$, 4d central charge $c_{\rm 4d}$, and associated BPS monodromy matrix $M$. For each $k$, define

$${\cal I}_k(q) \coloneq q^{\frac12 c(k)}(q;q)_\infty^{2r}\,{\rm Tr}\, M^k.$$

Here

$$c(k)=-k\,c_{\rm 4d}+\frac{r\,(k+1)}{6}.$$

Fixed-order MLDE conjecture. For any given 4d ${\cal N}=2$ superconformal field theory, ${\cal I}_k(q)$ solves a modular linear differential equation of fixed order and Wronskian index, for all $k$. This conjecture proposes a uniform modular-differential-equation structure for traces of powers of BPS monodromy matrices; the source motivates it by the observed generalized Schur partition functions and their fixed-order and fixed-Wronskian-index MLDEs, but gives no general proof.