Zlobin's Fibonacci basis conjecture for ordinary Euler sums

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

$$\left\{\zeta(\overline{b_1},b_2,\dots,b_d):d\geq1,\ b_j\in\{1,2\},\ b_1+\cdots+b_d=w\right\}.$$

Consequently, $\dim_{\mathbb Q}\mathsf{ES}_w=F_w$ for all $w\geq1$. The source presents this as a comparison conjecture; its general status is not specified.