Zhao's Fibonacci basis conjecture for finite Euler sums

Let $\mathsf{FES}_w$ be the $\mathbb{Q}$-vector space generated by finite Euler sums of weight $w$, and let $F_0=F_1=1$ and $F_k=F_{k-1}+F_{k-2}$ for $k\geq2$. Zhao's finite Euler sum basis conjecture. For every positive integer $w$, the following set is a basis of $\mathsf{FES}_w$:

$$\left\{\zeta_{\mathcal A}(\bar1,b_2,\dots,b_d):d\geq0,\ b_j\in\{1,2\},\ 1+b_2+\cdots+b_d=w\right\}.$$

Consequently, $\dim_{\mathbb Q}\mathsf{FES}_w=F_{w-1}$ for all $w\geq1$. The conjecture is supported by computations through weight six, while the general statement remains open.