The level-5 multiple-polylogarithm conjecture at the golden-ratio argument

Let $\operatorname{Li}_n(z)$ denote the polylogarithm of positive integer order $n$, let

$$\rho=\frac{\sqrt{5}-1}{2},$$

and let $\textsf{CMZV}^{5}_{n}$ denote the space of cyclotomic multiple zeta values of weight $n$ and level $5$. Golden-ratio level-5 conjecture. For every positive integer $n\geq 1$,

$$\operatorname{Li}_n(\rho^3)-\operatorname{Li}_n(-\rho^3)\in\textsf{CMZV}^{5}_{n}.$$

The supplied excerpt does not state verification or numerical evidence for this assertion, so its resolution remains to be checked.