Excentral $X_3$-inconic axis-ratio conjecture for 3-periodics

Let $I'_3$ be the $X_3$-centered inconic of the excentral triangle, and let $\mu'_3$ and $\mu_3$ denote its major and minor semi-axes, respectively. Let $a$ and $b$ be the semiaxes of the elliptic billiard, let $R$ and $r$ be the circumradius and inradius of the reference 3-periodic, let $d=\sqrt{R(R-2r)}$, and let $\rho=r/R$. Write $\delta$ for the parameter used in the source's formula.

Excentral $X_3$-inconic conjecture. The ratio $\mu'_3/\mu_3$ is invariant over all 3-periodics and is given by

$$\frac{\mu'_3}{\mu_3}=\frac{\sqrt{2\delta(\delta+a^2-b^2)-a^2b^2}}{b^2}=\frac{R+d}{R-d}=\frac{1+\sqrt{1-2\rho}}{\rho}-1.$$

The formula was derived for an isosceles 3-periodic and matches the ratio numerically for arbitrary combinations of $a$ and $b$; no proof is supplied.