This question is similar to this previous one but I think it is harder.

Let $X$, $Y$, $Z$, and $W$ be $2\times 2$ Hermitian matrices. Can we always find $\theta,\phi \in [0,\pi/2]$ and $2\times 2$ unitaries $U$ and $V$ such that $$SUXU^*S + CVYV^*C$$ and $$SUZU^*S + CVWV^*C,$$ are both scalar multiples of the identity matrix, where $S = {\rm diag}(\sin \theta, \sin \phi)$ and $C = {\rm diag}(\cos \theta, \cos\phi)$?

I'm guessing this is true. Maybe there is a homotopy reason as in the previous question. I think an answer in either direction would likely lead to a solution to this other previous question.

Edit: another way to ask this is, given $X$, $Y$, $Z$, and $W$, can we find $2\times 2$ matrices $A$ and $B$ such that $$AA^* + BB^*$$ $$AXA^* + BYB^*$$ $$AZA^* + BWB^*$$ are all scalar multiples of the identity (and the first one is not zero).

Edit 2: another reformulation. Let $\alpha$ and $\beta$ be the column vectors of $A^*$ and $\gamma$ and $\delta$ the column vectors of $B^*$ in the first edit. Then by looking at matrix entries of the expressions given there we see that the conditions become $$\langle X\alpha, \alpha\rangle + \langle Y\gamma, \gamma\rangle = \langle X\beta, \beta\rangle + \langle Y\delta, \delta\rangle$$ and $$\langle X\alpha,\beta\rangle + \langle Y\gamma,\delta\rangle = 0,$$ plus similar conditions with $Z, W$ or $I_2,I_2$ in place of $X,Y$. The problem is to find $\alpha,\beta,\gamma,\delta \in \mathbb{C}^2$, not all zero, satisfying these equations.