Superbity-two dimension conjecture for finite multiple zeta values

For each positive integer $w$, let $\FMZ_{w,2}$ be the $\Q$-vector space generated by finite multiple zeta values of weight $w$ and superbity two. Define $d_{0,2}=1$ and $d_{w,2}=\dim\FMZ_{w,2}$. Superbity-two dimension conjecture. One has $d_{1,2}=0$, $d_{2,2}=1$, and

$$d_{w,2}=d_{w-2,2}+d_{w-3,2}\qquad\text{for all }w\ge3.$$

The conjecture is supported by numerical calculations, including data for the initial weights; the source does not establish the recurrence in general.