The tight linear forest Turán conjecture for k congruent to 1 modulo r

Let $r\geq 2$, and let $\mathcal{L}_{n,k}^{(r)}$ be the family of all tight linear forests of order $n$ with $k$ edges in $r$-graphs. Here a tight linear forest is an $r$-graph whose connected components are tight paths or isolated vertices. Tight linear forest Turán conjecture. For $k>r$ and $k\equiv 1\pmod r$,

$$ex_r(n;\mathcal{L}_{n,k}^{(r)})=\max\left\{\binom{k+r-2}{r},\binom{n}{r}-\binom{n-(k-1)/r}{r}\right\}.$$

This conjecture gives an exact form of the Turán number for the family of tight linear forests when $k\equiv 1\pmod r$, strengthening the paper's asymptotic result in the stated dense regimes. The authors note that the corresponding error term should vanish in this congruence class; the conjecture is also proposed as a direct route to the Erdős Matching Conjecture.