Strongly regular graphs with Hoffman's condition

It is known that if the minimal eigenvalue of a graph is $-2$, then the graph satisfies Hoffman's condition; i.e., for any generated complete bipartite subgraph $K_{1,3}$ with parts $\{p\}$ and $\{q_1,q_2,q_3\}$, any vertex distinct from $p$ and adjacent to two vertices from the second part is not adjacent to the third vertex and is adjacent to $p$. We prove the converse statement, formulated for strongly regular graphs containing a 3-claw and satisfying the condition $gm>1$.