Parity conjecture for multiple Hurwitz polylogarithms

Let $r>1$, let $x_1,\ldots,x_r$ be roots of unity, let $a\in\mathbb{C}\setminus\mathbb{Z}$, and let $k_1,\ldots,k_r\geq1$ with $(k_r,x_r)\neq(1,1)$. Parity conjecture. The combination

$$\operatorname{Li}_{k_1,\ldots,k_r}(x_1,\ldots,x_r;a)+(-1)^{k_1+\cdots+k_r}x_1x_2\cdots x_r\operatorname{Li}_{k_1,\ldots,k_r}(x_1^{-1},\ldots,x_r^{-1};1-a)$$

can be expressed as a rational linear combination of multiple Hurwitz polylogarithm functions of depth less than $r$. This proposes a parity reduction for arbitrary depth; the source presents it as conjectural and notes that known cyclotomic multiple $t$-value conjectures give special cases.