Note on the mean value of the Erdős–Hooley Delta-function

For integer $n\geqslant 1$ and real $u$, let $\varDelta (n,u):=|\{d:d\,|\,n,\, \mathrm e ^u \lt d\leqslant \mathrm e ^{u+1}\}|$. The Erdős–Hooley Delta-function is then defined by $\varDelta (n):=\max_{u\in \mathbb R}\varDelta (n,u).$ We improve a recent upper bound for the mean value of this function by showing that, for large $x$, we have $$\sum _{n\leqslant x}\varDelta (n)\ll x(\log _2x)^{5/2}.$$