Maybe the answer to my question is obvious.

Let $p$ be a prime $\geq 3$. Let $D$ be an étale $(\varphi, \Gamma)$-module over $A_{\mathbb{Q}_p} = \{ \sum_{n \in \mathbb{Z}} a_n X^n \, \vert \, a_n \in \mathbb{Z}_p, a_n \to 0 \mbox{ as } n\to -\infty \}$ of finite rank.

Recall that the action of $\varphi$ is given by $\varphi(X) = (1+X)^p -1$ and that of $\gamma \in \Gamma$ by $\gamma(X) = (1+X)^{\omega(\gamma)} -1$ where $\omega$ is the $p$-adic cyclotomic character.

Denote by $A^+$ the subring of $A_{\mathbb{Q}_p}$ of power series without denominators, that is $A^+ = \{ \sum_{n \in \mathbb{N}} a_n X^n \, \vert \, a_n \in \mathbb{Z}_p \}$.

Say that $D$ is of finite height if $D= D^+ \otimes_{A^+} A_{\mathbb{Q}_p}$ where $D^+$ is an étale $(\varphi, \Gamma)$-module over $A^+$.

Is there an étale $(\varphi, \Gamma)$-module $D$ over $A_{\mathbb{Q}_p}$ (of finite rank) which is not of finite height and such that for all étale $(\varphi, \Gamma)$-module of rank $1$ $\delta$ over $A_{\mathbb{Q}_p}$, $D \otimes \delta$ is not of finite height ?

Obviously if such an object exists, it should be of rank $\geq 2$.