Edge-size conjecture for diameter-2, 3_t-critical graphs

Let $G$ be a simple graph of order $n$. A graph is $3_t$-critical if its total domination number is $3$ and adding an edge between any nonadjacent vertices lowers its total domination number. Suppose in addition that $G$ has diameter $2$, and let $m$ denote its number of edges. Edge-size conjecture for diameter-2, $3_t$-critical graphs. Then

$$m\geq\frac{n(n-2)}{4}.$$

A proof of this conjecture would, according to the source, imply the Murty–Simon conjecture together with the cited results for the diameter-3 case; the supplied text does not state that it has been resolved.