This question assumes familiarity with combinatorial cardinal characteristics of the continuum. It is abstracted out of a question in a joint research with Jialiang He. I hope we've got the abstraction right.

A family of subsets of $\mathbb{N}$ is *centered* if every finite subfamily
has an infinite intersection. A *pseudointersection* of a family is an infinite set that is almost contained in every member of the family.

Let $\mathfrak{ridiculous}$ be the the minimal cardinality of a centered family of subsets of $\mathbb{N}$ with no 2 to 1 image that has a pseudointersection.

By *2 to 1 image* of a family $\mathcal{A}$
we mean the family $\{f[A] : A\in\mathcal{A}\}$,
for some 2 to 1 function $f\colon \mathbb{N}\to \mathbb{N}$.
($f[A]:=\{f(n):n\in A\}$).

We know that $\mathfrak{p}\le\mathfrak{ridiculous}\le \operatorname{add}(\mathcal{M})$. (We see this using selection principles; direct arguments of course must exist, too.)

**Question.** Is $\mathfrak{ridiculous}=\mathfrak{p}$?

A negative answer (i.e., consistently ``no'') would be ridiculous.