Degree-four median-eigenvalue conjecture excluding projective-plane incidence graphs

Let $G$ be a simple graph of order $n$, with adjacency eigenvalues $\lambda_1 \geq \cdots \geq \lambda_n$, and let $\lambda_h$ and $\lambda_\ell$ be its median eigenvalues, where

$$h=\lfloor (n+1)/2\rfloor,\qquad \ell=\lceil (n+1)/2\rceil.$$

A vertex-disjoint union is a graph whose connected components are the specified graphs. Degree-four median-eigenvalue conjecture. If $G$ has maximum degree at most $4$ and is not a vertex-disjoint union of incidence graphs of projective planes of order $3$, then both median eigenvalues have absolute value at most $\sqrt{2}$. This is presented as a natural strengthening motivated by the resolved subcubic case and remains open.