Establish the consecutive-set property for minimum saturated families

Prove or disprove that, for every finite poset \(\mathcal P\) whose induced saturation number tends to infinity, every minimum-size \(\mathcal P\)-saturated family on \([n]\) contains, for every \(i\in[n]\), a pair of sets \(S,S\cup\{i\}\) with \(i\notin S\), once \(n\) is sufficiently large.

Background

The paper gives an example of a $\widehat{\mathcal V}$-saturated family lacking such consecutive pairs for some ground-set elements, but that example is not minimum-sized. Motivated by this distinction, the authors formulate a conjecture for minimum saturated families of arbitrary finite posets with unbounded saturation numbers. They note that the conjecture would imply the broader bounded-versus-linear dichotomy lower bound $\operatorname{sat}^*(n,\mathcal P)\geq n+1$.