Sparse triangle-free diameter-two graph conjecture

Let $G$ be a finite simple graph. A graph is triangle-free if it contains no triangle, and has diameter $2$ if every two vertices are at distance at most $2$. Sparse triangle-free diameter-two graph conjecture. For every integer $t\geqslant2$, there exists an integer $n_0$ such that if $G$ is a triangle-free diameter-$2$ graph that does not contain $K_{2,t}$ as a subgraph and has $n\geqslant n_0$ vertices, then $G$ is the star graph $K_{1,n-1}$. The conjecture is attributed in the paper to Wood and is presented as an open problem.