Equality conjecture for the hard edge tacnode temperature-derivative functions

Let $g_\sigma(s,x)$ and $h_\sigma(s,x)$ be the functions appearing in the rank-one temperature derivative of the hard edge tacnode kernel, namely

$$\frac{\partial}{\partial \sigma}K^\alpha(s,x;t,y)=y^\alpha g_\sigma(s,x)h_\sigma(-t,y).$$

Equality conjecture. The functions $g_\sigma$ and $h_\sigma$ are the same. The statement is presented without proof in the context of the temperature derivative theorem; the supplied text gives no further evidence that it has been established or refuted, so its general status remains open.