In this post, the answer notes that Lemma 3.4 gives:
Statement: Let ${\bf B}: [0, T]\times \Omega \rightarrow \mathbf{R}^p$ be the standard $p$-dimensional Brownian motion and ${\bf M}: [0, T]\times \Omega \rightarrow \mathbf{R}^{p\times p}$ be a matrix-valued stochastic process adapted to the natural filtration of the Brownian motion. Then $$\mathbf{E} \left[ \left\Vert\int_0^T {\bf M}_t d{\bf B}_t\right\Vert_2^2\right] = \mathbf{E} \left[ \int_0^T \left\Vert{\bf M}_t\right\Vert_F^2 dt\right]$$
However, in the paper there is no proof of the multidimensional version. I think I'm misunderstanding what exactly $\mathbf{M}$ is, and moreover can't figure out how to adapt the proof of Ito Isometry for the 1-dimensional case, which I can more-or-less follow from my class text. If I assume this result, then I can prove an exercise I'm working on, but from my naive understanding I should only be able to get a $\leq$ bound through triangle inequality rather than the true equality that I need and the statement promises.