Let $\sqsubset=\bigcup_n \sqsubset_n$ be a relation on $\omega^\omega$ where each $\sqsubset_n$ is arithmetic and $\{f: f\sqsubset_n g\}$ is closed for each $g\in \omega^\omega, n\in \omega, i.e. \Pi_1^0(g)$, a typical example is domination past $n$. In the following fix $\chi $ large enough regular cardinal.

Definition [Goldstern]. $\mathbb{Q}$ almost preserves $\sqsubset$ (or is almost preserving) if for all countable $N\prec H(\chi)$ containing $\mathbb{Q}, p\in \mathbb{Q}\cap N, g\in \omega^\omega$ and $N$ is $\sqsubset$-covered by $g$ (any $f\in N, f\sqsubset g$). There exists $q\leq p$ $(N,\mathbb{Q})$-generic such that $q\Vdash \forall f\in N[G] f\sqsubset g$.

We can check that this notion is preserved under iterations of finite length, more precisely, if $P$ is almost preserving and $1\Vdash_P \dot{Q}$ is almost preserving, then $P*\dot{Q}$ is almost preserving. My question is: is there any known example (references) that this notion is not preserved at limit stages (countable support iteration of proper forcings)?