Let $\sigma$ be a progressively measurable stochastic process on a probability space with a standard Brownian motion $W$. The Burkholder-David-Gundy inequality tells us that
$$ \mathbb E \Bigg( \sup_{s \le t \le s + \delta} \bigg| \int_s^t \sigma_u \mathrm dW_u \bigg|^p \Bigg) \le C_p \mathbb E \Bigg( \bigg( \int_s^t \sigma_u^2 \mathrm d u \bigg)^{p/2} \Bigg) $$
Do we also have something like $$ \mathbb E \Bigg( \sup_{s \le t \le s + \delta} \bigg| \int_s^t \sigma_u \mathrm dW_u \bigg|^p \Bigg| \mathcal F_s \Bigg) \le C_p \mathbb E \Bigg( \bigg( \int_s^t \sigma_u^2 \mathrm d u \bigg)^{p/2} \Bigg| \mathcal F_s \Bigg) $$ under any conditions?