Bounds conjecture for the Stirling error continued-fraction approximants

Let $r_n$ denote the error term in Stirling's formula, and let $g_m(n)$ be defined as in the paper from coefficients $a_m$ obtained from Theorem 3. The parity of $m$ determines the proposed direction of the bound.

Bounds conjecture. The functions $g_m(n)$ give lower bounds for $r_n$ when $m$ is even and upper bounds when $m$ is odd; equivalently,

$$r_n \begin{cases}\leq g_m(n), & m\text{ even};\\geq g_m(n), & m\text{ odd}.\end{cases}$$

The conjecture is based on numerical evidence and would establish the alternating upper and lower bounds suggested by the authors' telescoping continued-fraction construction. The supplied text gives no resolution, so its status remains open.