Question: For everyeven$k\ge 4$, is the following $(\star)$ true?$$\begin{align}\text{If $\beta=0.{a_1}^{k}{a_2}^{k}{a_3}^{k}\cdots\in\mathbb Q$, then $\alpha=0.a_1a_2a_3\cdots\in\mathbb Q$.}\qquad(\star)\end{align}$$

Here, $a_i$ is the $i$-th decimal place of $\alpha$.

**Example** : For $k=2,\alpha=0.12345,$ we have $\beta=0.1491625.$

**Motivation** : I've been able to prove that the answer is **NO** for $k=2$, and that the answer is **YES** for every **odd** $k\ge 3$. The answer seems YES for **any** $k\ge 3$, but I'm facing difficulty for treating every even $k$ in general. Can anyone help?

**Remark** : One of the counterexamples for $k=2$ is $\alpha=0.237237723777237777\cdots$. This question has been asked previously on math.SE without receiving any answers.