Cambie–Cames van Batenburg–Joannis de Verclos–Kang asymptotic conjecture for $h_t(\Delta)$

For positive integers $\Delta$ and $t$, let $h_t(\Delta)$ be the smallest integer such that every graph $G$ with at least $h_t(\Delta)$ edges and maximum degree $\Delta(G)\le \Delta$ contains two edges at distance at least $t$. Cambie–Cames van Batenburg–Joannis de Verclos–Kang conjecture. For $t\ge 3$ and every $\varepsilon>0$,

$$h_t(\Delta)\le (1+\varepsilon)\Delta^t$$

for all sufficiently large $\Delta$. This is an asymptotic strengthening of the known general upper bound of order $\frac32\Delta^t$. It remains open according to the supplied paper context.