Bound on zero-block lengths in the 2-adic expansion of zeta

Let $\zeta\in\mathbb{Z}_2$ be the $2$-adic number defined in Equation~, and let $\ell_\zeta(a)$ denote the length of the zero block beginning at the relevant digit position indexed by $a$. Zero-block length conjecture. For every $a\geq 0$,

$$\ell_\zeta(a)\leq\frac{2}{95}a+\frac{18}{5}.$$

By Proposition~, such a bound would control the growth rate of $\nu_2(U_i)$ through the approximability of $\zeta$ by non-negative integers. The source gives no proof or resolution of this assertion.