countable intersection of domain of densely defined operators is dense

Question

Let $(A,D(A))$ a closed densely defined linear operator on a Banach space $X$.

It is easy to prove that $D(A^n)$ is dense in $X$.

Define now $$D(A^\infty):=\bigcap_n^\infty D(A^n)$$ Is it true that $D(A^\infty)$ is dense in $X$?

I thought to use the Baire Cathegory theorem, but it works only for open dense sets and i don't know if the domain of the operator is open.