Exact Berge-path saturation conjecture

Let $\operatorname{sat}_k(n,\text{Berge-}P_m)$ denote the minimum number of edges in a $k$-uniform hypergraph on $n$ vertices that is Berge-$P_m$-saturated, and let $a^{(k)}_m$ be the parameter defined in the paper. Exact Berge-path saturation conjecture. For $k\geq 3$, $m\geq 10$, and $n\geq (k-1)a^{(k)}_m+1$,

$$\operatorname{sat}_k(n,\text{Berge-}P_m)=\left\lceil\frac{1}{k-1}\left(n-\left\lfloor\frac{n}{(k-1)a^{(k)}_m+1}\right\rfloor\right)\right\rceil.$$

The conjecture asserts that the upper and lower bounds proved in the paper coincide, equivalently that the minimal saturated construction has no isolated vertices. The paper establishes bounds differing by at most three, but does not resolve whether equality holds in the stated range.