Padé continued-fraction coefficient matching conjecture

Let $a_m$ be the sequence of numbers generated by Theorem 3, and let $g_m(n)$ be the associated finite continued fractions. Let $B(n)$ denote the continued fraction obtained from the Stirling series by the quotient-difference algorithm, with numerator sequence $(b_m)$.

Coefficient matching conjecture. In the limit as $m$ tends to infinity, the continued fraction $g_m(n)$ agrees with $B(n)$; equivalently, their numerator sequences satisfy

$$a_m=b_m.$$

This conjecture arises from the observed agreement between the coefficients produced by the telescoping method and those of the $S$-fraction for the diagonal Padé approximants to the Stirling series. No proof or disproof is supplied in the excerpt, so the conjecture remains open.