The asymptotic upper-bound conjecture for edge diameter

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 $t$. The parameter $t$ is fixed.

Asymptotic upper-bound conjecture. For $t\ne 2$ and any $\varepsilon>0$,

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

for all sufficiently large $\Delta$.

This is the complementary challenge to the asymptotic lower bound and would identify the asymptotic value of $h_t(\Delta)$ up to a factor tending to one. The paper states that the conjecture remains open for the relevant values of $t$, while the case $t=2$ is excluded because its known leading constant is $5/4$ rather than $1$.