Oyama's derivation relation conjecture for finite and symmetrized multiple zeta values

Let $\mathfrak H^1$ be the relevant noncommutative polynomial algebra of words, let $z=x+y$, and let $\partial_l$ be the derivation indexed by $l\in\mathbb Z_{\geq1}$. For $\mathcal F=\mathcal A$ or $\mathcal S$, let $Z_{\mathcal F}$ denote respectively the map assigning finite or symmetrized multiple zeta values to words. Oyama's conjecture. For every $l\in\mathbb Z_{\geq1}$ and every $w\in\mathfrak H^1$, one has

$$Z_{\mathcal F}(\partial_l(w))=-Z_{\mathcal F}(z^{l-1}yw),\qquad \mathcal F=\mathcal A\text{ or }\mathcal S.$$

These relations would give a unified family of derivation relations for both finite and symmetrized multiple zeta values. The source presents them as a conjecture, and no resolution is supplied here.