Cambie–Cames van Batenburg–Joannis de Verclos–Kang conjecture for $h_3(\Delta)$

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.

$$h_3(\Delta)\le \Delta^3-\Delta^2+\Delta+2.$$

This concerns the next case of the edge degree-diameter problem after the solved $t=2$ case. The paper states it as an open conjecture and notes that blow-ups of $C_5$ provide tight examples for the preceding case.