The level-4 multiple-polylogarithm conjecture

Let $\operatorname{Li}_n(z)$ denote the polylogarithm of positive integer order $n$, and let $i=\sqrt{-1}$. Let $\textsf{CMZV}^{2}_{n}$ denote the space of cyclotomic multiple zeta values of weight $n$ and level $2$. Level-4 conjecture. For every positive integer $n\geq 1$,

$$\operatorname{Li}_n\left(\frac{1+i}{2}\right)+\operatorname{Li}_n\left(\frac{1-i}{2}\right)\in\textsf{CMZV}^{2}_{n}.$$

This has been verified for $n\leq 6$, while numerical evidence exists for $n=7$. The verified cases require solving for all level-$4$ cyclotomic multiple zeta values of the relevant weight, whose number grows exponentially.