My question arises from the previous question
Let $M$ be a finitely generated $\mathbb{Q}[\mathbb{Z}/p^l]$-module, where $p$ is a prime number.
Is it true that \begin{equation}\dim_\mathbb{Q} M\geq \frac{p}{p-1}\dim_\mathbb{Q} M \mathbin{\mathop{\otimes}_{\mathbb{Q}[\mathbb{Z}/p^l]}}\mathbb{Q}(\zeta_{p^l})?\end{equation}
Here, $\zeta_{p^l}$ is the $p^l$th root of the unity.