Problem:
Given a random isotropic unit vector in $\mathbb{R}^p$ for $p\ge2$, we are trying to compute (preferably exactly, otherwise to upper bound): $$\mathbb{E}_{\mathbf{w}\sim\mathcal{S}^{p-1}}\!\left[{w_1}^{\!4}\,{w_2}^{\!4}\right]\,.$$

Any help would be greatly appreciated!

A related expectation we have already derived:
Using an answer to another question, we were able to compute $\mathbb{E}_{\mathbf{w}\sim\mathcal{S}^{p-1}}\!\left[{w_1}^{\!2}\,{w_2}^{\!2}\right]$ as follows: \begin{align}1 &=\mathbb{E} \left(\sum_{i=1}^p w_i^2\right)^2 \\[10pt] & = p(p-1)\cdot {\mathbb{E}_{\mathbf{w}\sim\mathcal{S}^{p-1}}\!\left[{w_{1}}^{\!2}\,{w_{2}}^{\!2}\right]}+p\cdot \mathbb{E} w_1^4 \\[10pt] {\mathbb{E}_{\mathbf{w}\sim\mathcal{S}^{p-1}}\!\left[{w_{1}}^{\!2}\,{w_{2}}^{\!2}\right]} & =\frac{1-p\cdot \mathbb{E} w_1^4}{p(p-1)}\stackrel{(*)}{=} \frac{1}{p\left(p+2\right)}\,,\end{align} where to show $(*)$ we notice that $w_{1}^{2}=\frac{X}{X+Y}$ where $X\sim\chi^{2}\left(1\right)=\Gamma\left(\frac{1}{2},\frac{1}{2}\right)$ and $Y\sim\chi^{2}\left(p-1\right)=\Gamma\left(\frac{p-1}{2},\frac{1}{2}\right)$, and conclude that $w_{1}^{2}=\frac{X}{X+Y}\sim B\left(\frac{1}{2},\frac{p-1}{2}\right)$ and $\mathbb{E}_{w_1}\left[w_1^4\right] =\operatorname{Var}\left[w_1^2 \right] + \left(\mathbb{E}_{w_1} \left[w_1^2 \right] \right)^2= \dots =\frac{3}{p(p+2)}$.