Coll–Mayers–Mayers conjecture on the seaweed index generating function

Let $G(q)$ be the $q$-series

$$G(q):=\prod_{n=1}^\infty \frac{1}{1+(-1)^nq^{2n-1}}=\left(q,-q^3;q^4\right)_\infty^{-1},$$

and let $e(n)$ and $o(n)$ denote respectively the numbers of partitions of $n$ into odd parts whose index is even and odd. Coll–Mayers–Mayers conjecture. The following statements hold: (1) all coefficients of $G(q)$ are non-negative; and (2)

$$G(q)=\sum_{n\geq 0}|e(n)-o(n)|q^n.$$

This conjecture connects the index statistic of seaweed algebras with a partition-theoretic $q$-series; the supplied source gives no evidence that it has been resolved.