Uniform boundedness of external saturation for posets

Let $P$ be a poset, and let $\mathop{}\\!\operatorname{extsat}(n,P)$ denote the minimum size of an external $P$-saturated family for the Boolean lattice ${\mathcal B}_n$, as defined in the paper. External saturation conjecture. For every poset $P$, there exists a constant $C_P$ independent of $n$ such that

$$\operatorname{extsat}(n,P)\le C_P.$$

The authors note that the dichotomy arguments for projective saturation also apply to external saturation, but no poset with unbounded external saturation number is known. This is the precise formal version of the preceding abstract-level conjecture.