If the Earth were to be flat, but still rotating with a constant angular frequency, would the Foucault pendulum still show it's precessing motion?

The angular velocity of the precession can be derived to be:
$$
\omega = -(\Omega \cos\beta)\mathbf{k}
$$
where $\Omega$ is the angular frequency of the earth, $\mathbf{k}$ is the apparent vertical and $\beta$ is the angle between the apparent vertical and the rotation axis.

On a rotating disk, with the same rotational frequency as the Earth, one would expect that the formula for $\omega$ would still be valid, albeit with a different apparent vertical $\mathbf{k}$.

**EDIT:** Gravity on the disk is assumed to be uniformly perpendicular to the disk, such as most flat-Earth theories suggest.

**N.B.** I do not believe the Earth to be flat, this question is purely out of interest.