The characterization conjecture for uniquely $C_{4}^{+}$-saturated graphs

Let $C_{4}^{+}$ be the diamond graph consisting of a $4$-cycle with one chord, and let $C_{3}^{*}$ be the graph consisting of a triangle with a pendant edge. A graph is strongly regular with parameters $(n,k,0,2)$ if it has $n$ vertices, is $k$-regular, every adjacent pair has no common neighbor, and every nonadjacent pair has exactly two common neighbors. A graph is nontrivial uniquely $C_{4}^{+}$-saturated if it is $C_{4}^{+}$-free and adding any missing edge creates exactly one copy of $C_{4}^{+}$, with at least four vertices.

Uniquely $C_{4}^{+}$-saturation conjecture. A graph $G$ is nontrivial uniquely $C_{4}^{+}$-saturated if and only if $G$ is a strongly regular graph with parameters $(n,k,0,2)$ or $G\cong C_{3}^{*}$.

The conjecture proposes a complete classification of these graphs. The source establishes the one-triangle case and rules out graphs with two, three, or four triangles, but leaves the possibility of more than four triangles unresolved.