Let $\Sigma=\{0,1\}^{\mathbb Z}$ and let $\sigma:\Sigma\to\Sigma$ be the left shift. Then it is well known that $(\Sigma, \sigma)$ is conjugate to the baker's map $B$ of the unit square: $$ B(x,y) = \begin{cases} (2x, \frac{y}{2}) &\text{ if } 0 \leq x < \frac{1}{2}, \\ (2x-1, \frac{y+1}{2}) &\text{ if } \frac{1}{2} \leq x < 1. \end{cases} $$ via the conjugating map $\pi:\Sigma\to[0,1]^2$ defined as follows: $$ \pi((x_n)_{-\infty}^\infty)=\left(\sum_{n=1}^\infty x_n2^{-n}, \sum_{n=0}^\infty x_{-n}2^{-n-1} \right). $$ Now assume $X$ to be a subshift of $\Sigma$ and let $h$ stand for topological entropy.

**Question.** Is it true that
$$
\dim_H(\pi(X))=\frac{2h(X)}{\log 2}?
$$
A weaker version (which is still sufficient for my needs) is whether $h(X)=0$ is equivalent to $\dim_H(\pi(X))=0$.

This looks like a standard result; however, the usual settings are one-dimensional or maps on manifolds.