Symmetry conjecture for multiple Hurwitz polylogarithms

Let $x_1,\ldots,x_r$ be roots of unity and let $a\in\mathbb{C}\setminus\mathbb{Z}$. For $\boldsymbol{k}=(k_1,\ldots,k_r)\in\mathbb{N}^r$ with $(k_1,x_1)\neq(1,1)$ and $(k_r,x_r)\neq(1,1)$, define congruence modulo products by discarding all product terms of cyclotomic multiple Hurwitz zeta values of depth less than $r$. Symmetry conjecture.

$$\operatorname{Li}_{k_1,\ldots,k_r}(x_1,\ldots,x_r;a)\equiv(-1)^{k_1+\cdots+k_r-1}(x_1\cdots x_r)\operatorname{Li}_{k_r,\ldots,k_1}(x_r^{-1},\ldots,x_1^{-1};1-a)\pmod{\text{products}}.$$

The source presents this as the counterpart of the parity conjecture; its general validity remains open.