  For polynomial ideals in positive charachteristic, defining $F$-split rings
and admitting a squarefree monomial initial ideal are different notions. In
this note we show that, however, there are strong interactions in both
directions. Moreover we provide an overview on which $F$-singularities are
Gr\"obner deforming. Also, we prove the following characteristic-free
statement: if $\mathfrak{p}$ is a height $h$ prime ideal such that
$\textrm{in}(\mathfrak{p}^{(h)})$ contains at least one squarefree monomial,
then $\textrm{in}(\mathfrak{p})$ is a squarefree monomial ideal.
