Let $(W_t)_{t\ge 0}$ be a standard Brownian motion. For each $t\in [0,1]$, it is known that, e.g. from Burkholder-Davis-Gundy's inequality

$$\mathbb E\big[\sup_{s\in [t,t+\Delta t]}|W_s-W_t|^p\big]=O(\Delta t^{p/2}),\quad \forall p\ge 1,$$

where $O$ refers to "of order". Do we have an estimate of

$$\mathbb E\big[\sup_{s,t\in[0,1],~ |s-t|<\Delta t}|W_s-W_t|^p\big]?$$

A student asked me this question when I taught the explicit Euler's scheme applied to SDEs, but I cannot find any reference. Any answers, comments or references are appreciated.