The conjecture extending the sign pattern of $h_1$ to every $h_n$

Let $d$ be the integer parameter in the transfer-operator construction, let $q>0$, and let $h_n$ denote the functions introduced in the paper. For $k=1,\ldots,d$, define

$$Q_k=\begin{cases} \big(2(k-1),2k\big) & \text{if } k=1,\ldots,d-1,\\ \big(2(d-1),\infty\big) & \text{if } k=d.\end{cases}$$

Sign-pattern conjecture. Lemma (b) is true for every $h_n$, $n\in\mathbb{N}$; that is, for $q\in Q_k$ and $0<t<1/2$, one has $(-1)^{k-1}h_n'(t)>0$.

This is proposed on the basis of computer experiments as an extension of the proved statement for $h_1$.