Extremal formula for Berge-disjoint paths

Let $\ell,r$ be integers and let $k\geq 2$. Write

$$\ell'=\left\lfloor\frac{\ell+1}{2}\right\rfloor.$$

For an $r$-uniform hypergraph, let $\mathrm{ex}_r(n,\text{Berge-} kP_\ell)$ denote the maximum number of hyperedges in an $n$-vertex hypergraph containing no Berge copy of $k$ vertex-disjoint paths of length $\ell$. The extremal formula for Berge-disjoint paths. There exists $N_{\ell,r,k}$ such that, whenever $n>N_{\ell,r,k}$ and $2\leq r\leq k\left\lfloor\frac{\ell+1}{2}\right\rfloor$,

$$\mathrm{ex}_r(n,\text{Berge-} kP_\ell)=\binom{k\ell'-1}{r-1}(n-k\ell'-1+1)+\binom{k\ell'-1}{r}+\mathbb{I}_r\binom{k\ell'-1}{r-2}.$$

Here $\mathbb{I}_r$ is the indicator appearing in the stated formula. The conjecture proposes that the proved extremal result remains valid throughout the indicated wider range of $r$; the source notes that the lower bound follows from a similar construction, while the upper bound remains to be established.