Exact connected Turán number conjecture for Berge paths

Let $r$ and $k$ be integers satisfying

$$r+1\le k\le 2r-1.$$

For an $r$-uniform hypergraph on $n$ vertices, let $\operatorname{ex}^{\mathrm{conn}}_r(n,\mathcal{B}P_k)$ denote the maximum number of edges in a connected hypergraph containing no Berge path of length $k$. Exact connected Turán number conjecture. For sufficiently large $n$,

$$\operatorname{ex}^{\mathrm{conn}}_r(n,\mathcal{B}P_k)=n-(k-2)+\binom{k-2}{r}.$$

The paper determines the connected Turán number in some parameter ranges and gives asymptotic results for $k=r+1$; this proposed formula extends the expected behavior to the full range $r+1\le k\le 2r-1$, but its resolution is not supplied.