The direct-sum conjecture for special values of Witten multiple zeta values attached to $\mathfrak{sl}_4$

Let $\MZV(w,\le l)$ be the $\Q$-vector space generated by multiple zeta values of weight $w$ and depth at most $l$, and let $\gz_\slfour$ denote the Witten multiple zeta function attached to $\mathfrak{sl}_4$. For $w>3$, consider its special values at nonnegative integers. Direct-sum conjecture. The space generated over $\Q$ by these special values is

$$\MZV(w,\le 3)\oplus \MZV(w-1,\le 2)\oplus \MZV(w-2,1).$$

The preceding theorem gives inclusion in the corresponding sum, while the conjecture asserts that this sum is direct.