The asymptotic eigenvalue conjecture for the generalized Gauss-Kuzmin-Wirsing operator

Let $d4d8_p$ be the operator on functions defined by

$$\mathscr{H}_pf(x)=\frac{p}{(p+x)^2}f\left(\frac{p}{p+x}\right).$$

For every positive integer $p$, let $d4d8_p$ denote the $n$-th eigenvalue of the generalized Gauss-Kuzmin-Wirsing operator and define $d719_p$ by

$$\phi_p=\frac{\sqrt{p^2+4p}-p}{2},$$

so that $d719_p$ is the positive root of $x=\frac{p}{p+x}$. Asymptotic eigenvalue conjecture. For every positive integer $p$,

$$\lim_{n\to\infty}\frac{\Lambda_p(n)}{(-1)^{n-1}p^{-n}\phi_p^{2n}}=1.$$

This conjecture generalizes the known asymptotic description for the case $p=1$, attributed in the source to G. Alkauskas. It predicts the leading asymptotic behavior, including the alternating sign and exponential scale, of the eigenvalues for every positive integer parameter $p$.