The cubic edge diameter conjecture for maximum degree

Let $h_t(\Delta)$ be the smallest integer such that every graph with $h_t(\Delta)$ edges and maximum degree at most $\Delta$ contains two edges whose distance is at least $t$; equivalently, $h_t(\Delta)-1$ is the largest number of edges in a graph of maximum degree at most $\Delta$ whose line graph has diameter at most $t$. Here $h_3(\Delta)$ denotes the case $t=3$.

Cubic edge diameter conjecture.

$$h_3(\Delta)\leq \Delta^3-\Delta^2+\Delta+2,$$

with equality if $\Delta$ is one more than a prime power.

The bound is motivated by point-line incidence graphs of finite projective planes, which attain $\Delta^3-\Delta^2+\Delta$ edges before a one-edge subdivision. The conjecture has been confirmed for $\Delta=3$, where $h_3(3)=23$, but remains open in general.