$3 rightarrow 1$ QRAC encoding for XOR functions

I found a way to do it for the $2to1$ QRAC. I simply guessed that we could leave the measurement bases as they are, $Z$ for the first bit, and $X$ for the second bit, and added $Y$ as the basis with which to extract the XOR of the two bits. From the guess obtaining the optimal encoding states is then easy, we just need to diagonalize the relevant operators. They are begin{align*} |psi_{00}rangle & = sqrt{p}|0rangle + sqrt{1-p} e^{frac{pi i}4}|1rangle |psi_{01}rangle & = sqrt{p}|0rangle + sqrt{1-p} e^{-frac{3pi i }4}|1rangle |psi_{10}rangle & = sqrt{1-p}|0rangle + sqrt{p} e^{-frac{pi i}4}|1rangle |psi_{11}rangle & = sqrt{1-p}|0rangle + sqrt{p} e^{frac{3pi i}4}|1rangle, end{align*} where $p=frac{3+sqrt3}{6}$ is also the probability of success in this QRAC. I don't know whether it is optimal, but I guess it is, since it's so nice and symmetrical.

Now for the $3to1$ case such a simply idea cannot work, as you want to extract 7 different functions, but there are only these 3 mutually unbiased bases in dimension 2. What I would do is to find some set of 7 bases that is in some sense uniformly distributed on the Bloch sphere and see if it works. Another idea is to simply do a see-saw optimization and see what you get.