Famously, a countable support iteration with proper iterands is again proper. This is mostly stated as follows: Suppose $(\mathbb{P}_{\alpha},\dot{\mathbb{Q}}_{\alpha})_{\alpha<\lambda}$ is a countable support iteration such that for each $\alpha<\lambda$, $\mathbb{P}_{\alpha}\Vdash$ "$\dot{\mathbb{Q}}_{\alpha}$ is proper". Then the countable support limit $\mathbb{P}_{\lambda}$ is proper. I am interested in what happens if we merely require each $\mathbb{P}_{\alpha}$ to be proper (note that $\mathbb{P}_{\alpha}*\dot{\mathbb{Q}}_{\alpha}$ being proper does not imply that $\mathbb{P}_{\alpha}$ forces $\dot{\mathbb{Q}}_{\alpha}$ to be proper by Properness of quotient forcing). The question could probably be resolved by answering the following:

Suppose $(\mathbb{P}_n,\dot{\mathbb{Q}}_n)_{n\in\omega}$ is an iteration such that $\mathbb{P}_n$ is proper for each $n\in\omega$. Is the fully supported limit $\mathbb{P}_{\omega}$ proper as well?