Tyomkyn–Uzzell's balanced double-broom conjecture for triangle-free distance graphs

Let $G$ be an $n$-vertex graph, and let $G_k$ be its graph of pairs at distance exactly $k$. Write $D_k$ for the $k$-distance graph of a graph $D$, and call $D$ a balanced double broom when it is the double broom specified in the conjecture.

Tyomkyn–Uzzell's conjecture. For $k\geq 3$ and $n\geq k+1$, except when $k=3$ and $n=8$, every $n$-vertex graph $G$ such that $G_k$ is triangle-free satisfies

$$|E(G_k)|\leq\frac{(n-k+1)^2}{4}.$$

Moreover, if equality holds, then $G_k$ is isomorphic to $D_k$, where $D$ is a balanced double broom.

This conjecture strengthens the previously known result for sufficiently large $k$ and sufficiently large $n$ as a function of $k$, extending it to all indicated values of $k$ and $n$. The source says that it would imply the corresponding conclusion for $k=3$ and $n\geq 9$; no resolution is supplied here.