Padovan dimension conjecture for finite multiple zeta values of arbitrary superbity

For positive integers $w$ and $\ell$, let $\FMZ_{w,\ell}$ be the $\Q$-vector space generated by finite multiple zeta values of weight $w$ and superbity $\ell$. Arbitrary-superbity dimension conjecture.

$$\FMZ_{w,\ell}\cong\frac{\MZV_w}{\zeta(2)\MZV_{w-2}}\oplus\frac{\MZV_{w+1}}{\zeta(2)\MZV_{w-1}}\oplus\cdots\oplus\frac{\MZV_{w+\ell-1}}{\zeta(2)\MZV_{w+\ell-3}}.$$

This conjecture extends the proposed Kaneko–Zagier correspondence from superbity one to arbitrary superbity and is motivated by numerical dimension data.