Odd-index reduction conjecture for the series H

For positive integers $b_1,\dots,b_s$, let $H_{b_1,\dots,b_s}(z)$ denote the series used in the paper, and let $H_{a_1,\dots,a_r}(z)$ be the corresponding series with indices $a_i$. Odd-index reduction conjecture. Every series $H_{b_1,\dots,b_s}(z)$ can be expressed as a linear combination over $\mathbb Z[z,z^{-1}]$ of $1$ and series $H_{a_1,\dots,a_r}(z)$ such that all $a_i$ are odd, $r\le s$, and

$$a_1+\cdots+a_r\le b_1+\cdots+b_s.$$

This is presented as a strengthening suggested by computer computations of an earlier lemma. The source supplies no proof, resolution, or counterexample.