Padovan dimension conjecture for finite multiple zeta values of superbity one

For each positive integer $w$, let $\FMZ_w$ be the $\Q$-vector space generated by finite multiple zeta values of weight $w$ and superbity one. Define $d_0=1$ and $d_w=\dim\FMZ_w$ for $w\ge1$. Superbity-one dimension conjecture. One has $d_1=d_2=0$, and

$$d_w=d_{w-2}+d_{w-3}\qquad\forall w\ge3.$$

Moreover, for $w\ge3$, a basis is

$$\{\zeta_{\calA_1}(1,2,a_1,\dots,a_r):a_1,\dots,a_r\in\{2,3\},\ a_1+\cdots+a_r=w-3\},$$

and all $\Q$-linear relations among finite multiple zeta values are generated by the double-shuffle relations. The conjecture is motivated by numerical evidence and by the analogous dimension conjecture for ordinary multiple zeta values.