Conjecture that LPS graphs are lossless expanders

Let $q$ be fixed and large, and let $X_n$ be the family of $(q+1)$-regular Ramanujan graphs constructed by Lubotzky, Phillips, and Sarnak. For a vertex set $Y\subset X_n$, write $N(Y)$ for its neighborhood. LPS lossless-expansion conjecture. For every $\epsilon>0$, there is $n$ large enough such that, for every set $Y\subset X_n$ with

$$|Y|\leq |X_n|^{1-\epsilon},$$

we have

$$|N(Y)|\geq (q+1-o(q))|Y|.$$

Lossless expansion is the optimal vertex-expansion behavior expected of these number-theoretic Ramanujan graphs. The paper's method applies to Morgenstern graphs rather than LPS graphs and suggests that the absence of suitable closed subgroups may lead to good expansion, but the conjecture is presented as unproved.