Uniqueness of odd-order girth-eight algebraic graphs

Determine whether every graph BΓ3(Fq; f2, f3) with f2,f3∈Fq[p1,l1] and girth at least eight is isomorphic to BΓ3(Fq; p1l1,p1l1^2), and determine whether every graph BΓ2(Fq; f2,f3) with f2∈Fq[p1,l1] and f3∈Fq[p1,l1,p2,l2] and girth at least eight is isomorphic to BΓ2(Fq; p1l1,p1l1^2), for odd prime powers q.

Background

The paper surveys uniqueness theorems showing that broad classes of algebraically defined graphs of girth at least eight are isomorphic to the standard biaffine generalized-quadrangle graph. It then poses two broader finite-field questions; a negative answer could produce new generalized quadrangles.