Setting: Consider sampling two orthonormal vectors $\mathbf{u},\mathbf{v}$ in $\mathbb{R}^p$ (where $p\ge2$) from a "uniform" distribution over the $p$-dimensional sphere (alternatively, sample $\mathbf{u},\mathbf{v}\sim\mathcal{S}^{p-1}$ and then orthogonalize and normalize $\mathbf{v}$).

Let $\lceil p/2\rceil\le m\le p$.
Question: Can we upper bound the following expectation? $$ \mathbb{E}_{\substack{\mathbf{u},\mathbf{v}\sim\mathcal{S}^{p-1}:\\\mathbf{u}\perp\mathbf{v}}} \left(\mathbf{u}^\top\left[\begin{array}{cc} \mathbf{I}_{m}\\ & \mathbf{0}_{p-m} \end{array}\right]\mathbf{v}\right)^2 = \mathbb{E}_{\substack{\mathbf{u},\mathbf{v}\sim\mathcal{S}^{p-1}:\\\mathbf{u}\perp\mathbf{v}}} \left(\sum_{i=1}^{m}u_iv_i\right)^2 $$

Example: when $m$ is very close to $p$, the expectation should be very small, since the vectors are nearly orthogonal.

Direction: We thought about approximating this with just two independent Gaussian vectors and look for the concentration bounds when $p\to\infty$, but this seems perhaps too loose?

Observation: Since the vectors are orthogonal, we have $ \left(\mathbf{u}^\top\left[\begin{array}{cc} \mathbf{I}_{m}\\ & \mathbf{0}_{p-m} \end{array}\right]\mathbf{v}\right)^2 = \left(\mathbf{u}^\top\mathbf{v}-\mathbf{u}^\top\left[\begin{array}{cc} \mathbf{I}_{m}\\ & \mathbf{0}_{p-m} \end{array}\right]\mathbf{v}\right)^2 = \left(\mathbf{u}^\top\left[\begin{array}{cc} \mathbf{0}_{m}\\ & \mathbf{I}_{p-m} \end{array}\right]\mathbf{v}\right)^2$, and thus we can equivalently focus on $0\le m\le \lfloor p/2\rfloor$.

Any help and ideas would be greatly appreciated!