A *tower* is a subset $T\subset [\omega]^\omega$ of the family $[\omega]^\omega$ of all infinite subsets of $\omega$ such that $T$ is well-ordered by the relation $\supset^*$ of almost inclusion and has no infinite pseudointersections. A tower is *regular* if its cardinality is a regular cardinal.

Consider two small uncountable cardinals:

$\mathfrak t=\min\{|T|:T\subset[\omega]^\omega$ is a tower$\}$;

$\hat{\mathfrak t}=\sup\{|T|:T\subset[\omega]^\omega$ is a regular tower$\}$.

It is clear that $$\mathfrak t\le\hat{\mathfrak t}\le\mathfrak c.$$ Martin's Axiom implies $\mathfrak t=\hat{\mathfrak t}=\mathfrak c$. On the other hand the strict inequality $\mathfrak t<\hat{\mathfrak t}$ is known to be consistent (but I do not know the reference).

The cardinal $\mathfrak t$, called the *tower number*, is well-studied in Set Theory.

I am interested in the cardinal $\hat{\mathfrak t}$, more precisely in its relation to other known cardinal characteristics of the continuum.

Problem 1.Is there any non-trivial upper or lower bound for the cardinal $\hat{\mathfrak t}$?

Problem 2.In which known models of ZFC does the strict inequality $\mathfrak t<\hat{\mathfrak t}$ hold?

Maybe there is some fixed standard notation for $\hat{\mathfrak t}$?