Hefetz–Keevash's Turán conjecture for extensions of two-edge matchings

Let $r\geq 4$ and let $H_{2r}^{M_2^r}$ be the extension of the $r$-uniform matching $M_2^r$ consisting of two pairwise disjoint edges. Let $G$ be an $r$-graph on $[n]$, and let $S^r(n)$ be the $r$-graph whose vertex set is partitioned into $A$ and $B$, whose edges have one vertex in $A$ and $r-1$ vertices in $B$, with the part sizes chosen to maximise the number of edges. Hefetz–Keevash's conjecture. For sufficiently large $n$,

$$ex(n,H_{2r}^{M_2^r})={1 \over r}n\cdot{{r-1 \over r}n \choose r-1}.$$

Moreover, if $G$ is $H_{2r}^{M_2^r}$-free and has this many edges, then $G\cong S^r(n)$. This conjecture predicts both the exact Turán number and the unique extremal construction for the extension of a two-edge matching; the supplied text gives no resolution status.