Determine the exact induced saturation number of the isolated-point fork

Determine whether the exact induced saturation number of the four-point poset consisting of a three-point fork \(\mathcal V\) together with an isolated point, denoted \(\widehat{\mathcal V}\), is \(n+2\) for all sufficiently large \(n\), or establish the precise value of \(\operatorname{sat}^*(n,\widehat{\mathcal V})\).

Background

The paper proves the bounds $$n/2+2\leq \operatorname{sat}^*(n,\widehat{\mathcal V})\leq n+2$$. The authors explain that improving the lower-bound counting argument by eliminating collisions among witness sets would yield the exact value $n+2$, and state that they strongly believe this improvement is possible. The proposed problem is therefore to resolve whether the upper bound is exact.