Modalized modal sentence

A modal sentence $A$ is said to be modalized in $p$ if every occurrence of $p$ happens within the scope of a $\square$.

As an example, $\square p\wedge q$ is modalized in $p$, but not in $q$.

If $A$ does not contain $p$, then it is trivially modalized in $p$.

A sentence which is modalized in every sentence letter is said to be fully modalized.

Being modalized in $p$ is a sufficient condition for having a fixed point on $p$.