Broadhurst-type dimension conjecture for finite Euler sums of superbity one

Let $\FES_{w,1}$ denote the $\Q$-vector space of finite Euler sums of weight $w$ and superbity one, and set $f_w=\dim_{\Q}\FES_{w,1}$ for $w\ge1$. Finite Euler-sum dimension conjecture.

$$\sum_{w=1}^{\infty}f_wt^w=\frac{t}{1-t-t^2}.$$

Moreover, $\ES_{w,1}$ has the basis

$$\{\zeta_{\calA_1}(\bar1,a_1,\dots,a_r):a_1,\dots,a_r\in\{1,2\},\ a_1+\cdots+a_r=w-1\}.$$

This is the finite-Euler-sum analogue of the dimension conjectures for finite multiple zeta values; the source gives numerical motivation but no proof.