Amply 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: for any generated complete bipartite subgraph $K_{1,3}$ (a 3-claw) with parts $\{p\}$ and $\{q_1, q_2,q_3\}$, any vertex distinct from $p$ and adjacent to the vertices $q_1$ and $q_2$ is adjacent to $p$ but not adjacent to $q_3$. We prove the converse statement for amply regular graphs containing a 3-claw and satisfying the condition $\mu>1$.