Depth-three symmetry conjecture for multiple zeta-value series

Let $n\geq 0$ be an integer. Let $\sigma\in\{2,3\}$ be the element satisfying $\sigma\equiv n\pmod 2$, and let $\widetilde{\sigma}$ be the other element of $\{2,3\}$. Let $P(k_1,k_2,k_3)$ be a polynomial with rational coefficients satisfying

$$\begin{cases} P(-k_1-5n-4,k_2,k_3)=-P(k_1,k_2,k_3),\\ P(k_1,-k_2-3n-2,k_3)=P(k_1,k_2,k_3),\\ P(k_1,k_2,-k_3-n)=-P(k_1,k_2,k_3), \end{cases}$$

and $\deg_{k_i}P\leq 3n+1$ for each $i\in\{1,2,3\}$. Depth-three symmetry conjecture. The series

$$\sum_{k_1\geq k_2\geq k_3\geq 1}\frac{P(k_1,k_2,k_3)}{(k_1+2n+2)_{n+1}^{3}(k_2+n+1)_{n+1}^{3}(k_3)_{n+1}^{3}}$$

is a rational linear combination of $1$, $\zeta(2)$, $\zeta(3)$, $\zeta(2,3)$, $\zeta(3,2)$, $\zeta(3,3)$, and $\zeta(\sigma,\widetilde{\sigma},\sigma)$. The source reports verification for $n\leq 2$ and proposes this as a conjecture for general $n$; it concerns control of the depth-$2$ part of such series.