Quantitative convergence conjecture for partial Stirling functions

Let $x\in\mathbb Z_2$ be a finite element, and let $i_0$ and $d$ be positive integers such that $x_{i_0}=0$ and $x_{i+d}=x_i$ for all $i\ge i_0$, with $2^{i+d}\le x$. Define

$$R(x):=\operatorname{lg}(x)+1-(i_0+d),$$

the number of repeating bits of $x$. The quantitative convergence conjecture. Provided $R(x)\ge2(e-i_0)-1$, one has

$$\nu\bigl(P_{2^{e+d}+1}(x+1)-P_{2^e+1}(x+1)\bigr)\ge e-i_0.$$

This gives an explicit lower bound on the 2-adic closeness of successive terms in the periodic subsequences.