A coefficient conjecture for a family of double Lambert series

Let $a$ be a positive integer, let $n$ be a positive integer, and write $[q^k]F(q)$ for the coefficient of $q^k$ in $F(q)$. Let

$$\sigma_1(n)=\sum_{d\mid n}d$$

be the sum of the divisors of $n$. Coefficient conjecture. One has

$$[q^{n2^a}]\sum_{m,n\geq1}\frac{q^{mn2^a}}{(1+q^{n2^{a-1}})(1-q^{2m-1})}=\sigma_1(n).$$

This conjecture predicts divisor-sum coefficients for a family of double Lambert series indexed by $a$. The source presents it as one of two conjectures and does not provide evidence of a resolution.