Gewirtz graph as an induced subgraph of triangular graphs

Determine whether the Gewirtz graph, the strongly regular graph with parameters $(n',d',\lambda',\mu')=(56,10,0,2)$, can occur as an induced subgraph of the triangular graph $T_\ell=L(K_\ell)$ for any integer $\ell\geq 32.

Background

The paper studies necessary conditions for a strongly regular graph to occur as an induced subgraph of another strongly regular graph, using both the Lovász $\vartheta$-function and graph energy. For triangular graphs $T_\ell$, these conditions reduce to two explicit inequalities involving the parameters of a candidate induced strongly regular subgraph.

Applying these inequalities to the Gewirtz graph, which has 56 vertices and strongly regular parameters $(56,10,0,2)$, rules out its occurrence as an induced subgraph of $T_\ell$ for every $\ell\leq31$. For $\ell\geq32$, both necessary inequalities are satisfied, so the paper leaves unresolved whether an actual induced-subgraph embedding exists.