On the friable mean-value of the Erdős-Hooley Delta function

For integer $n$ and real $u$, define $\mathrm{\Delta }(n,u):=\mathrm{\mid }\{d:d\mid n,{\mathrm{e}}^u<d⩽{\mathrm{e}}^{u+1}\}\mathrm{\mid }$. Then, put $\mathrm{\Delta }(n):={\max }_{u\in \mathbb{R}}\mathrm{\Delta }(n,u)\mathrm{.}$ We provide uniform upper and lower bounds for the mean-value of $\mathrm{\Delta }(n)$ over friable integers, i.e. integers free of large prime factors.