Determine the general MLDE order for the T_m theories

Determine whether the modular linear differential equation for the Schur index of the four-dimensional theories T_m=(A_2,D_{3m+1}) has order 3m+2 for every positive integer m.

Background

The paper studies the proposed correspondence between the Argyres–Douglas theories T_m=(A_2,D_{3m+1}) and the doublet VOAs A(4m+2). For m=2, the authors explicitly derive an order-8 modular linear differential equation for the modularly normalized vacuum character/index, and they report that the analogous order formula has been verified for m=1, 2, and 3.

The unresolved problem is to establish the conjectured pattern for arbitrary m, thereby determining the general modular differential equation governing the associated Schur indices or VOA characters.