Graphs with projective suborbits

A positive answer is given to the problem of the existence of an upper bound for orders of stabilizers of vertices of connected graphs in vertex-transitive groups of automorphisms, if these stabilizers are finite and, under actions on adjacent vertices, contain the group $\mathbf{PSL}_n(q)$ as normal subgroups in its natural doubly transitive representation, $\operatorname{char}\mathbf F_q>3$ ($n$ and $q$ fixed).