Hoffman's conjecture for symmetric triple series

Let $A=3$ and let $n\geq 0$ be an integer. Define

$$K_1=k_1+\frac{5n}{2}+2,\qquad K_2=k_2+\frac{3n}{2}+1,\qquad K_3=k_3+\frac{n}{2}.$$

For a polynomial $P(K_1,K_2,K_3)$ in any of the four families specified in Theorem

$$of the source, with the stated parity conditions on its non-negative exponents, consider the series$$

\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}^{A}(k_2+n+1){n+1}^{A}(k_3)_{n+1}^{A}}.

$$**Hoffman's symmetric-series conjecture.** Whenever this series is convergent, Theorem$$

holds: it is a rational linear combination of multiple zeta values of depth at most $3$, with neither $\zeta(2,2)$ nor $\zeta(2,2,2)$ appearing; and, if the additional condition $(8)$ holds, it is a rational linear combination of the explicitly listed values $1$, $\zeta(2)$, $\zeta(3)$, $\zeta(2,3)$, $\zeta(3,2)$, $\zeta(3,3)$, $\zeta(2,2,3)$, $\zeta(2,3,2)$, $\zeta(2,3,3)$, $\zeta(3,2,2)$, $\zeta(3,2,3)$, $\zeta(3,3,2)$, and $\zeta(3,3,3)$. The source says this was checked computationally for $n\leq 2$ and leaves the general case open; in particular, the open problem is to prove that $\zeta(2,2)$ does not appear.