Conjecture on the limiting probability that a random lift is Ramanujan

Let $G$ be a Ramanujan graph, and let $H$ be its random uniform $n$-lift. A graph is Ramanujan when all its nontrivial eigenvalues have absolute value at most the Ramanujan bound $2\sqrt{d-1}$ for a $d$-regular graph.

Ramanujan-lift probability conjecture. There exists a constant $c$ with $0<c<1$ such that

$$\mathbb{P}(H\text{ is Ramanujan})=c+o(1),$$

where the $o(1)$ term tends to $0$ as $n\to\infty$. This predicts that the probability of the Ramanujan property for random lifts converges to a nontrivial limit, rather than to either zero or one. The source presents this as a natural conjecture motivated by the unresolved probability that a random regular graph is Ramanujan; no resolution status is given.