$p$-sequentiality and $p$-Fréchet-Urysohn property of Franklin compact spaces

Franklin compact spaces defined by maximal almost disjoint families of subsets of $\omega$ are considered from the view of its $p$-sequentiality and $p$-Fréchet-Urysohn-property for ultrafilters $p\in \omega^*$. Our principal results are the following: CH implies that for every $P$-point $p\in \omega^*$ there are a Franklin compact $p$-Fréchet-Urysohn space and a Franklin compact space which is not $p$-Fréchet-Urysohn; and, assuming CH, for every Franklin compact space there is a $P$-point $q\in \omega^*$ such that it is not $q$-Fréchet-Urysohn. Some new problems are raised.