Given two set $A,B$ we write $A\subset^* B$ if the complement $A\setminus B$ is infinite.

A *Hausdorff gap* is a transfinite family $\langle A_\alpha,B_\alpha\rangle_{\alpha\in\omega_1}$ of infinite subsets of $\omega$ satisfying the following two properties:

$\bullet$ $A_\alpha\subset^* A_\beta\subset^* B_\beta\subset^* B_\alpha$ for any countable ordinals $\alpha\le\beta$;

$\bullet$ for any infinite subset $I\subset \omega$ there exists a countable ordinal $\alpha$ such that $A_\alpha\not\subset^* I$ or $I\not\subset^* B_\alpha$.

A in infinite subset $I\subset\omega$ is called a *pseudointersection* of a Hausdorff gap $\langle A_\alpha,B_\alpha\rangle_{\alpha\in\omega_1}$ if $I\subset^* (B_\alpha\setminus A_\alpha)$ for any $\alpha\in\omega_1$.

The definition of the small uncountable cardinal $\mathfrak p$ implies that under $\mathfrak p>\omega_1$ each Hausdorff gap has an infinite pseudointersection.

What does happen under $\mathfrak p=\omega_1$?

Problem.Is $\mathfrak p=\omega_1$ equivalent to the existence of a Hausdorff gap without infinite pseudointersection?