Subsets of $\omega$-words which share certain factors and languages accepted by special (prefix-closed) automata
The conjecture does not hold:
Let $L$ be the set of prefixes of $(c^*ac^*b)^*$.
Then $L'=(c^*ac^*b)^\omega+ (c^*ac^*b)^*c^\omega+(c^*ac^*b)^*c^*ac^\omega$.
Take the word $\eta=(c^1ac^1b)(c^2ac^2b)(c^3ac^3b)\dots \in L'$.
For all $k\in\mathbb N$, we can show that $PF_k(\eta)\not\subseteq L'$, as witnessed by $$u_k=(c^1ac^1b)(c^2ac^2b)\dots (c^kac^kb)c^kb c^\omega.$$
Indeed we have $u_k\in PF_k(\eta)$ but $u_k\notin L'$.