Convergence conjecture for eventually periodic 2-adic arguments

Let $x\in\mathbb Z_2$ have binary digits $x_i\in\{0,1\}$, so that $x=\sum_{i\ge0}x_i2^i$. Suppose that, for some $d\ge2$ and $i_0\ge0$, $x_{i+d}=x_i$ for all $i\ge i_0$. The eventual-periodicity convergence conjecture. For every $e$, the limit

$$\lim_{j\to\infty}P_{2^{e+dj}+1}(x)$$

exists in $\mathbb Z_2$. This asserts convergence along the residue classes of exponents determined by the eventual binary period, extending the already established convergence for positive integer arguments.