Let $\Sigma_2 = \{0,1\}^{\mathbb{Z}}$ be the full two-shift with left-shift map $\sigma$ and the standard product metric $$d(x,y) = 2^{-\inf\{|n| : x_n \neq y_n\}}.$$
Fix $\varepsilon = 2^{-m}$ for some $m \in \mathbb{N}$. An $\varepsilon$-ball $B_\varepsilon(z)$ is the cylinder set determined by the word $z_{[-m,m]}$.
Given $x \in \Sigma_2$ and a finite set of times $0 \leq n_1 < \cdots < n_k$, we can create an edited sequence by choosing symbols $s_j \in \{0,1\}$ and setting $$\tilde{x}_n = \begin{cases} s_j & \text{if } n = n_j,\\ x_n & \text{otherwise}. \end{cases}$$
Applying $\sigma$ to this edited sequence gives an $\varepsilon$-pseudo-orbit with at most $k$ "jumps" (edits). Define $$k_\varepsilon(x \to z) := \min\bigl\{k : \exists\ \text{edited }\varepsilon\text{-pseudo-orbit of }x \text{ that is }\varepsilon\text{-shadowed by some }y\text{ with }\sigma^{N}(y)\in B_\varepsilon(z)\bigr\}.$$
Assume $x_i = z_i$ for all $i < 0$.
Question: For the full two-shift, what non-trivial bounds (beyond the obvious ceiling $k_\varepsilon(x \to z) \leq 2m+1$) exist for $k_\varepsilon(x \to z)$ in terms of the one-sided Hamming distance $$d_H(x_{[0,\infty)}, z_{[0,\infty)}) = \sum_{n=0}^{\infty} \mathbf{1}_{x_n \neq z_n}?$$
In particular, does there exist a universal constant $C(\varepsilon) < 2m+1$ such that $$k_\varepsilon(x \to z) \leq C(\varepsilon) \log\bigl(1 + d_H(x_{[0,n]}, z_{[0,n]})\bigr)$$ for all sufficiently large $n$?
