The Macdonald index formula for $(A_1,D_{2n+1})$ theories

Let $n$ be a positive integer, and let $f^n_{m,l}(q)$ be polynomials in $q$ satisfying

$$1+\sum_{m=1}^{\infty}q^m\sum_{l=1}^{m}\frac{f^n_{m,l}(q)}{(q;q)_l}(q^N;q)_l(q^{1-N};q)_l=q^{nN(N-1)}.$$

The $(A_1,D_{2n+1})$ Macdonald-index conjecture. The Macdonald index is

$${{\cal I}}^{(A_1,D_{2n+1})}_{\mathsf{M}}=1+\sum_{m=1}^{\infty}t^m\sum_{l=1}^{m}\frac{f^n_{m,l}(q)}{(q;q)_l}\sum_{k=0}^{2l}\binom{2l}{k}_q.$$

This proposes a uniform formula for the family of Argyres–Douglas theories and is motivated by arc-space computations and consistency with the Schur limit. The source supplies no proof or resolution.