The general equality conjecture for $H(m,\alpha)$ and $H_2(m,\alpha)$

Let $H(m,\alpha)$ and $H_2(m,\alpha)$ be the quantities considered in the paper, with $m\geq 3$ and $\alpha\geq 2$, and define

$$\lambda(\alpha)=\sum_{j=1}^{\infty}\frac{2^{j-1}}{\alpha^{2^j}}.$$

General equality conjecture.

$$H(m,\alpha)=H_{2}(m,\alpha)\quad\text{for all }m\geq 3\text{ and }\alpha\geq 2.$$

In particular,

$$\lim_{m\to\infty}\frac{2H(m,\alpha)}{m\alpha^m}=\lim_{m\to\infty}\frac{2H_{2}(m,\alpha)}{m\alpha^m}=\lambda(\alpha).$$

This generalizes the conjectured equality at $\alpha=2$. The paper proves matching structural and asymptotic results for $H_2(m,\alpha)$, while the equality with $H(m,\alpha)$ is left open.