Difference sets satisfying the Deza Cayley graph parameter condition

Determine whether there exist difference sets in abelian groups, other than the known examples, with parameters $(v,k,\lambda)$ satisfying $k=\frac{2v-1-\sqrt{8v-7}}{2}$, so that the associated generalized dihedral Cayley graph is a strictly Deza graph.

Background

Proposition \ref{ds} characterizes when a nontrivial difference set $R$ in an abelian group $A$ yields a Deza Cayley graph $\operatorname{Cay}(G,Rb\cup A^\#)$ over the generalized dihedral group $G=A\rtimes\langle b\rangle$. The necessary and sufficient condition is $k=\frac{2v-1-\sqrt{8v-7}}{2}$, and in that case the graph is strictly Deza.

The paper exhibits examples with parameters $(7,3,1)$, $(11,6,3)$, $(16,10,6)$, and $(37,28,21)$, producing several strictly Deza Cayley graphs. It then reports that database verification found no additional examples with $v\leq 10000$, leaving open whether any other difference sets satisfy the required condition.