Pseudodual grids and extensions of generalized quadrangles

We prove that an amply regular, locally $GQ(s,t)$ graph with $\mu=2t+4$ either has $s=t=2$, or is a Taylor graph, or has $s=2$ and $t=4$ and is a unique strongly regular, locally $GQ(2,4)$ graph with parameters (64, 27, 10, 12).