Andrews et al.'s fermionic formula conjecture for the Macdonald index

Let $k$ be a positive integer, and let $\mathcal{I}_{\mathsf{Mac}}^{(2k+1)}(z,t;q)$ denote the Macdonald index of the $(A_1,D_{2k+1})$ Argyres--Douglas theory. For $n\geq 0$, write $(a;q)_n$ for the $q$-Pochhammer symbol, and define the $q$-binomial coefficient by

$${M\brack N}_q=\begin{cases}\dfrac{(q;q)_M}{(q;q)_N(q;q)_{M-N}},&0\leq N\leq M,\\0,&\text{otherwise}.\end{cases}$$

Andrews et al.'s conjecture. The specialization of the Macdonald index at $z=1$ satisfies

$$\mathcal{I}_{\mathsf{Mac}}^{(2k+1)}(1,t;q)=\sum_{n_k\geq\cdots\geq n_1\geq0}\frac{t^{n_1+\cdots+n_k}q^{n_1^2+\cdots+n_{k-1}^2}}{(q;q)_{n_k-n_{k-1}}\cdots(q;q)_{n_2-n_1}(q;q)_{n_1}}\sum_{j=0}^{2n_k}{2n_k\brack j}_q.$$

This conjecture proposes a fermionic multisum representation for the Macdonald index, complementing the known bosonic sum-like expression. Its status is unresolved in the supplied source context.