Simplicity, sign, and ratio conjecture for the Gauss transfer operator eigenvalues

Let $\lambda_n$ be the eigenvalues of the transfer operator for the Gauss continued fraction map, arranged so that $|\lambda_1|\geq |\lambda_2|\geq\cdots$, and let $\phi=(1+\sqrt{5})/2$. Simplicity, sign, and ratio conjecture. The following three statements are true: the eigenvalues are simple and $|\lambda_n|$ strictly decreases; they have alternating sign,

$$(-1)^{n+1}\lambda_n>0;$$

and

$$\lim_{n\to\infty}\frac{\lambda_n}{\lambda_{n+1}}=-\frac{3+\sqrt{5}}{2}=-\phi^2.$$

These were described as three outstanding unresolved problems concerning the eigenvalues; the claims concern their simplicity, signs, and asymptotic ratio.