The set of $e \in \{0,1\}^{\mathbf{N}}$ such that $e_1+\cdots+e_n\neq o(n)$

Question

Endow the space $\{0,1\}^{\mathbf{N}}$ of sequences $(e_1,e_2,\ldots)$ with the product topology (of the discrete topology on $\{0,1\}$), which is completely metrizable. Modulo my mistakes, it has been proved here that the following set is meager: $$ \left\{(e_1,e_2,\ldots) \in \{0,1\}^{\mathbf{N}}: \liminf_{n\to \infty}\frac{e_1+\cdots+e_n}{n}>0\right\}. $$

Here we ask a stronger question (which probably has a negative answer):

Question. Is the following set meager? $$ \left\{(e_1,e_2,\ldots) \in \{0,1\}^{\mathbf{N}}: \limsup_{n\to \infty}\frac{e_1+\cdots+e_n}{n}>0\right\} $$

Answer 1

Flipping all the digits is a homeomorphism, so your first statement shows that $$\left\{(e_1,e_2,\ldots) \in \{0,1\}^{\mathbf{N}}: \limsup_{n\to \infty}\frac{e_1+\cdots+e_n}{n}<1\right\}$$ is meagre, so its complement is comeagre, and you're asking about an even bigger set.