Let $d\ge 3$ be a constant. Is there an explicit construction of an infinite family of $d$-regular graphs such that for $G$ in this family with $n$ vertices, every subgraph $H$ of on at most $\alpha n$ vertices where $\alpha$ is a constant depending on $d$ has average degree bounded by, say, $2.1$?

Note that a random $d$-regular graph satisfies this property (see, for example, Section 4.6 of https://www.cs.huji.ac.il/~nati/PAPERS/expander_survey.pdf for a proof).

Let's also say we aren't picky about achieving this for literally every choice of $d$, and would be happy with obtaining a construction for arbitrarily large $d$. The notion of explicit here is that the graph is constructible in polynomial time in the number of vertices. Let's also say we are happy if the average degree bound of $2.1$ is replaced by any constant $C$ as long as it does not depend on $d$.

Do any of the explicit constructions of Ramanujan graphs, such as the Lubotzky-Phillips-Sarnak/Margulis/Morgenstern constructions, achieve this property? The best bound we know on the average degree is about $\sqrt{d-1}+1$, which only uses the fact that the second largest magnitude eigenvalue is at most $2\sqrt{d-1}$. Is this bound improvable?