Graphs in which local subgraphs are strongly regular with second eigenvalue 5

J. Koolen proposed the problem of studying distance-regular graphs in which the neighborhoods of vertices are strongly regular graphs with second eigenvalue $\le t$ for a given positive integer $t$. Earlier Koolen's problem was solved for $t=4$. We complete the classification of distance-regular graphs in which the neighborhoods of vertices are strongly regular graphs with second eigenvalue $r$, where $4$<$r\le5$.