Fibonacci dimension conjecture for generalized polylogarithm values

For each weight $w$, let $\mathcal{L}_w$ be the $\mathbb{Q}$-linear space generated by the values $\mathop{\rm Li}\nolimits_{\vec{s}}(1/2)$ with $w(\vec{s})=w$, and let $f_0=f_1=1$ and $f_w=f_{w-2}+f_{w-1}$. Fibonacci dimension conjecture. For $w\geqslant1$, one has

$$\mathbb{F}_w=f_w,$$

where $\mathbb{F}_w=\dim_{\mathbb{Q}}\mathcal{L}_w$. This predicts Fibonacci growth for the dimensions of the spaces generated by these special values.