(This question assumes familiarity with combinatorial cardinal characteristics of the continnum.)

Let $[\mathbb{N}]^\infty$ be the family of infinite subsets of $\mathbb{N}$, partially ordered by $\subseteq^*$, where $a\subseteq^* b$ means $a\setminus b$ is finite.

Let $\mathfrak{ht}$ (sometimes called $\operatorname{Depth}^+([\mathbb{N}]^\infty)$) be the minimal cardinal number $\kappa$ such that there is no $\subset^*$-decreasing $\kappa$-sequence in $[\mathbb{N}]^\infty$. Then $\mathfrak{t}<\mathfrak{ht}\le\mathfrak{c}^+$.

A classic result from Kunen's thesis asserts that, when adding $\kappa\ge\aleph_1$ Cohen reals to a model of CH, we obtain $\mathfrak{ht}=\aleph_2$.

**Open-ended question.** Can the hypothesis $\mathfrak{b}<\mathfrak{ht}$ be expressed using cardinal characteristics other than $\mathfrak{ht}$?

It is easy to see that "$\mathfrak{t}=\mathfrak{b}$ or $\mathfrak{b}<\mathfrak{d}$" implies "$\mathfrak{b}<\mathfrak{ht}$".

**Question 1.** Is it consistent that
"$\aleph_1=\mathfrak{t}<\mathfrak{b}=\mathfrak{c}=\aleph_2<\mathfrak{ht}$"?

In the Laver model, we have $\aleph_1=\mathfrak{t}<\mathfrak{ht}=\mathfrak{b}=\mathfrak{c}=\aleph_2$.

**Question 2.** What is the value of $\mathfrak{ht}$ in the Hechler and Mathias models?

**Update:** Will Brian answers Question 1 in the positive below, in the Hechler model (thus also answering Question 2 for Hechler).
I thought that $\mathfrak{b}<\mathfrak{ht}$ implies there is a nontrivial set of reals satisfying the selection principle $\operatorname{S}_1(\Gamma,\Gamma)$, by
the the linked paper. But Brian's comments make it clear that I oversimplified the question for this purpose. I opened a new question that fits better the intended application.