The polylogarithmic expression conjecture for twisted partial zeta values

Let $k$ be as in the preceding theorem, let $calpha,cbeta$ be the parameters in the narrow ideal-class partial zeta value $czeta(k,(calpha,cbeta),cscrB)$, and let $cscrB^*$ denote the associated dual narrow ideal class. For $r\in\mathbb{N}$, consider the polylogarithms $\operatorname{Li}_{2r}(e^{2\pi i\alpha})$ and $\operatorname{Li}_{2r}(e^{2\pi i\beta})$. Polylogarithmic expression conjecture. The expression

$$\zeta(k,(\alpha,\beta),\mathscr{B})+(-1)^k\zeta(k,(\beta,\alpha),\mathscr{B}^*)$$

is a linear combination of

$$\operatorname{Li}_{2r}(e^{2\pi i\alpha})\quad\text{and}\quad \operatorname{Li}_{2r}(e^{2\pi i\beta}),\qquad r\in\mathbb{N}.$$

The conjecture would provide the desired generalization of the preceding theorem to unequal parameters $(\alpha,\beta)$ without introducing the functions $\mathscr{F}_k$ into the formula. The source does not state a resolution, so the conjecture is open.