Factored square-biroot identity at one

For a positive expansion parameter $c$, let $\beta_m^2(1,c)$ denote the factored square biroot evaluated at $1$. Factored square-biroot identity. The value is

$$\beta_m^2(1,c)=\frac{(c+1)^m+(c-1)^m}{(c+1)^m-(c-1)^m}.$$

The source presents this as an interesting but unproved observation that may help analyze the error at $1$ as the expansion parameter increases in the square-root case. It is therefore a speculative auxiliary conjecture rather than one of the paper’s main convergence results.