A two-term relation conjecture for depth-two multizeta values

Let $q$ be a power of $2$, and let $0\leq i\leq2$ and $0\leq j\leq2-i$. Define

$$b=q^3-iq^2-jq-(q-i-j),\qquad b'=q^3-iq^2-(q-(2-j))q-(2-i-j).$$

For $a\in\{4q,q^2+q-1,q^2+q,q^2+2q-2,q^2+2q-1,q^2+2q\}$, the two-term relation conjecture asserts

$$\zeta(a,b)=\frac{[1]^{(q-2)q}}{[2]^{q-2}}\zeta(a+b-b',b').$$

This proposes additional rational-function-field relations between depth-two multizeta values of equal weight. The source presents it as a conjecture and gives no evidence of resolution.