The conjecture that infinite Ramanujan graphs are not Liouville

Let $G$ be an infinite $d$-regular graph. Call $G$ Ramanujan if the spectral radius of its Markov operator on $l^2$ of the vertices equals

$$\frac{2\sqrt{d-1}}{d},$$

the spectral radius of the $d$-regular tree. For connected $G$, this spectral radius is also the limiting exponent of the return probabilities, $\lim p_n(v,v)^{1/n}$. The Ramanujan non-Liouville conjecture. Every infinite Ramanujan graph is not Liouville. The source cites partial affirmative results but does not state that the conjecture is resolved.