Extremal formula for Berge linear forests

Let $\mathcal{P}=P_{\ell_1}\cup\dots\cup P_{\ell_k}$ be a linear forest with $k\geq 2$ and $\ell_k\geq\dots\geq\ell_1\geq 3$. Write

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

For an $r$-uniform hypergraph, let $\mathrm{ex}_r(n,\text{Berge-}\mathcal{P})$ denote the maximum number of hyperedges in an $n$-vertex hypergraph containing no Berge copy of $\mathcal{P}$. Extremal formula for Berge linear forests. When $2\leq r\leq\sum_{i=1}^k\ell_i'$,

$$\mathrm{ex}_r(n,\text{Berge-}\mathcal{P})=\max\{\mathrm{ex}_r(n,\text{Berge-}P_{\ell_k}),f(n,\ell_i,r)\},$$

where

$$f(n,\ell_i,r)=\binom{\sum_{i=1}^k\ell_i'-1}{r-1}\left(n-\sum_{i=1}^k\ell_i'+1\right)+\binom{\sum_{i=1}^k\ell_i'-1}{r}+\mathbb{I}_{\Pi\ell_i}\cdot\binom{\sum_{i=1}^k\ell_i-1}{r-2},$$

and $\mathbb{I}_{\Pi\ell_i}=0$ if $\prod_{i=1}^k\ell_i$ is odd, while $\mathbb{I}_{\Pi\ell_i}=1$ if $\prod_{i=1}^k\ell_i$ is even. The conjecture extends the single-path extremal formula to arbitrary linear forests; the source says that a similar construction gives the lower bound, but does not provide a proof of the asserted equality.