Given a particular $1600$-bit output $Y$, is it feasible to find a $1600$-bit input $X$ such that $\text{Keccak-}f[1600](X) = Y$?

I have read this answer on Crypto.SE and I want to clarify a few details.

Does this property of $$\text{Keccak-}f[1600](x)$$ imply that if we know a particular $$1600$$-bit output $$Y$$, it is feasible to find a $$1600$$-bit input $$X$$ such that $$\text{Keccak-}f[1600](X) = Y$$?

If yes, then consider the following situation.

1. An adversary $$A$$ has access to the full $$1600$$-bit state before the final truncation.
2. The original message $$M$$ has the length $$L$$ equal to, say, $$552$$ bits, and $$A$$ knows the exact value of $$L$$.

In this situation, will $$A$$ be able to find $$M$$?