On locally $GQ(s,t)$ graphs with strongly regular $\mu$-subgraphs

The connected locally $GQ(s,t)$ graphs are studied in which every $\mu$-subgraph is a known strongly regular graph (i.e., $K_{m,m}$ for a positive integer $m$, the Moore graph with parameters $(k^2+1,k,0,1)$, $k=2,3$, or $7$, the Clebsch graph, the Gewirtz graph, the Higman-Sims graph, or the second neighborhood (with parameters $(77,16,0,4))$ of a vertex in the Higman-Sims graph). It is proved that if $\Gamma$ is a strongly regular locally $GQ(s,t)$ graph in which every $\mu$-subgraph is isomorphic to a known strongly regular graph $\Delta$, then one of the following statements is true: $(1)$ $\Delta=K_{t+1,t+1}$ and either $s=1$ and $\Gamma=K_{3\times (t+1)}$, or $s=4,t=1$, and $\Gamma$ is a quotient of the Johnson graph $\overline{J}(10,5)$, or $s=t=1,2,3,8,13$; $(2)$ $\Delta$ is a Petersen graph and $\Gamma$ is a unique locally $GQ(2,2)$ graph with parameters $(28,15,6,10)$; $(3)$ $\Delta$ is the Gewirtz graph and $\Gamma$ is the McLaughlin graph.