Is this known about $\pi$?

Question

Suppose that we travel along the digits of $\pi$ and when we arrive at $2$ or $3$ or $4$ or $5$ or $6$ or $7$ or $8$ or $9$ we replace those digits with the number $1$ and when we arrive at $1$ or $0$ we leave them unchanged. When we do this with all the digits will resulting number be irrational?

Since this looks quite hard you can, if you want, assume that $\pi$ is normal in order to prove this. But I do not know is normality enough?

(The statement that we can suppose that $\pi$ is normal was not in the original version of the question.)

Answer 1

So here is a question.

In a normal number, can the positions of the $0$s in the decimal be eventually periodic?

Answer: NO. In a normal number there are arbitrarily long strings of $7$s. (And also infinitely many $0$s.)