No covering with nowhere dense $\mathsf{\text{P}}$-sets in the Cohen model

We prove that if less than ${\mathrm{\aleph }}_{\omega }$-many Cohen reals are added to a model of $\mathsf{\text{CH}}$, then ${\omega }^{\ast }$ can not be covered by nowhere dense $\mathsf{\text{P}}$-sets (equivalently, there is an ultrafilter on $\omega$ that does not contain a tower).