The level-two finite multiple zeta value conjecture for finite Euler sums

Let $\mathcal{Z}_{\mathcal{A}}^{(2)}$ and $\mathcal{E}$ be the $\mathbb{Q}$-vector spaces of level-two finite multiple zeta values and finite Euler sums, respectively. For an index $(k_1,\ldots,k_r)$, let $\zeta_{\mathcal{A}}^{(2)}(k_1,\ldots,k_r)$ denote the corresponding level-two finite multiple zeta value. Level-two finite multiple zeta value conjecture.

$$\mathcal{Z}_{\mathcal{A}}^{(2)}=\mathcal{E}.$$

Moreover, the set

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

forms a linear basis of $\mathcal{Z}_{\mathcal{A}}^{(2)}$. This is supported by numerical experiments; the conjectural dimension of the weight-$k$ part is the Fibonacci number $F_k$, matching the cardinality of the proposed basis in weight $k$.