I am interested in the distribution of the parity of $\pi(x)$, the prime counting function, over the natural numbers.

Let:

$\ \ E_n := \left\{ k \in \left\{1,\dots,n\right\} : \pi(k) \equiv 0 \mod 2 \right\}\ \ $ and $\ \ O_n :=\left\{ k \in \left\{1,\ldots,n\right\} : \pi(k) \equiv 1 \mod 2 \right\}$.

My question: is it true that

$$\lim_{n \to \infty} \frac{|E_n|}{n} = \lim_{n \to \infty} \frac{|O_n|}{n} = \frac{1}{2}$$

Are any methods known for calculating or estimating these limits?