Let $\{W_t\}_{t\in[0;T]}$ be a one-dimensional Brownian motion and let $\{\mathcal F_t\}_{t\in[0;T]}$ be the augmented filtration generated by this Brownian motion. Let $\{\sigma_t\}_{t\in[0;T]}$ be real-valued, progressively measurable with $\mathbb E(\int^T_0 \sigma_t^2 \mathrm dt) < \infty$ and $\int^T_0 \sigma_t \mathrm dW_t \in \mathcal L^\infty$.
Is there a sequence of continuous and adapted processes $\{\tilde\sigma^n_t\}_{t\in[0;T]}$ such that $$ \lim_{n\to \infty} \mathbb E\Bigg( \int^T_0 \bigg( \int^t_0 \tilde\sigma^n_s \mathrm ds - \int^t_0 \sigma_s \mathrm dW_s \bigg)^2 \mathrm dt \Bigg) = 0? $$
If $\sigma$ is essentially bounded, the answer is yes (see this question). But what if $\sigma$ is unbounded?
By Burkholder-Davis-Gundy inequality, the condition $\int^T_0 \sigma_t \mathrm dW_t \in \mathcal L^\infty$ additionally implies $\mathbb E((\int^T_0 \sigma_t^2 \mathrm dt)^2) < \infty$.
However, I don't see whether we can obtain that $\sigma$ is essentially bounded. If not, I neither see whether Lemma 1 from this answer to the mentioned other question still holds, i.e. whether, for all $\beta,\delta \in (0;\infty)$, there exists $\nu \in (0;\infty)$ such that $$ \mathbb P \bigg[ \sup_{0 \le s \le t \le \min(s+\nu,T)} \bigg| \int^t_s \sigma_u \mathrm dW_u \bigg| \le \beta \bigg] \ge 1 - \delta. $$ If we had this property, the proof of the above limit property would become significantly easier reachable.
