For varieties $X,Y$ over an algebraically closed field, and a surjective morphism $f:X\rightarrow Y$, $\dim f^{-1}(y)\geq\dim X-\dim Y$ for all closed $y\in Y$, and $\dim f^{-1}(y)=\dim X-\dim Y$ for all closed $y$ in a nonempty open subset of $Y$. If the requirement that $X$ and $Y$ are of finite type is dropped, and we just require them to be integral, separated, Noetherian schemes over an algebraically closed field, there are two ways I can see to interpret this claim:

(1) $\dim f^{-1}(y)\#\dim Y\geq\dim X$ for all closed $y\in Y$, and $\dim f^{-1}(y)\#\dim Y=\dim X$ for all closed $y$ in a nonempty open subset of $Y$, where $\dim$ refers to ordinal Krull dimension, and $\#$ is natural sum ($\alpha\#\beta$ is the largest ordinal that can be reached by interleaving $\alpha$ and $\beta$).

(2) $\text{codim}(f^{-1}(y)\text{ in }X)\geq\dim Y$ for all closed $y\in Y$, and $\text{codim}(f^{-1}(y)\text{ in }X)=\dim Y$ for all closed $y$ in a nonempty open subset of $Y$, where $\text{codim}(Z\text{ in }X)$ refers to the supremum of order types of chains of irreducible closed subsets of $X$ containing $Z$.

These two claims are not clearly equivalent in the infinite-dimensional setting. Are either or both of them true?

Edit: Friedrich Knop has shown that (1) is false, but his counterexample convinced me that I didn't ask the question I intended to. I am still interested in knowing whether this is true when $f$ sends constructible sets to constructible sets, whether it is true when there is an integral, separated, Noetherian scheme $Z$ such that $X\subseteq Z^{n+m}$ is constructible and $f$ is the projection map $X\rightarrow Y\subseteq Z^n$, and whether this example implies the condition that $f$ sends constructible sets to constructible sets.