It is quite well known that

Any FS (finite support) iteration of length $<\mathfrak{c}^+$ of $\sigma$-centered posets is $\sigma$-centered (see e.g. here).

Now consider the following question: whenever $\delta<\mathfrak{c}^+$ is a limit ordinal of uncountable cofinality and $\langle\mathbb{P}_{\alpha}:\alpha<\delta\rangle$ is a sequence of $\sigma$-centered posets such that

(i) $\mathbb{P}_\alpha$ is a complete subposet of $\mathbb{P}_\beta$ for all $\alpha<\beta<\delta$, and

(ii) $\mathbb{P}_\gamma=\bigcup_{\alpha<\gamma}\mathbb{P}_\alpha$ for any limit $\gamma<\delta$,

do we have that $\mathbb{P}_\delta:=\bigcup_{\alpha<\delta}\mathbb{P}_\alpha$ is $\sigma$-centered? Is this correct in the case $\delta=\omega_1$?

Note that this may not be a consequence of the result mentioned at the beginning since the quotient of two $\sigma$-centered posets is not necessarily (forced to be) $\sigma$-centered.

Any help with this problem would be highly appreciated.