This is a sort-of follow up to this question, which I asked before I became confused about if things were reduced.

More specifically, suppose $\varphi:X\to Y$ is a surjective morphism of finite presentation between algebraic varieties (reduced, irreducible, separated schemes, finite type over an algebraically closed field), and furthermore assume that $Y\cong \mathbb{A}^r$ is an affine space.

I am able to show that the underlying reduced schemes of all the fibers $X_y$ for $y\in Y$ are smooth and all equidimensional of dimension $k>0$ (In my case $k=r$, so $\dim(X)=2r$), and also that each fiber is generically reduced. I even know that the generic fiber is reduced, so I know $X_y$ is reduced for $y\in U\subset Y$ in some nonempty open.

Can I conclude that the fibers are reduced?

More generally, under relatively simple circumstances, one can get that fibers over a Zariski-open are reduced. Is there some simple criteria as to when this may be strengthened to all fibers? (short of assuming $\varphi$ is smooth, for example.)

EDIT: As pointed out by Snowball, $X$ being Cohen-Macaulay would suffice. Perhaps a better formulation of the question is

Is $X$ necessarily Cohen-Macaulay, and if not are their known counterexamples?

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