The odd-index basis conjecture for ordinary finite multiple zeta values

Let $\mathcal{Z}_{\mathcal{A}}^{(1)}$ be the $\mathbb{Q}$-vector space of ordinary finite multiple zeta values, and let $\zeta_{\mathcal{A}}^{(2)}(k_1,\ldots,k_r)$ denote a level-two finite multiple zeta value. Odd-index basis conjecture. If all $k_i$ are greater than $1$, then

$$\zeta_{\mathcal{A}}^{(2)}(k_1,\ldots,k_r)\in\mathcal{Z}_{\mathcal{A}}^{(1)}.$$

Furthermore, the set

$$\{\zeta_{\mathcal{A}}^{(2)}(k_1,\ldots,k_r)\mid r\geq 1,\ k_i\geq 3\text{ odd for all }i\}$$

constitutes a basis of $\mathcal{Z}_{\mathcal{A}}^{(1)}$. The conjecture is presented as a further conclusion suggested by numerical experiments and refines the inclusion of ordinary finite multiple zeta values into the level-two space.