Conjecture on 2k-vertex-2-regular-free uniform hypergraphs

Let $k\geq 3$ be an integer. An $n$-vertex $k$-uniform hypergraph is said to have no $2$-regular subgraphs on $2k$ vertices if it contains no subhypergraph on exactly $2k$ vertices in which every vertex has degree $2$. The $2k$-vertex conjecture. There exists $n_k$ such that, for every $n\geq n_k$, every $n$-vertex $k$-uniform hypergraph $H$ with no $2$-regular subgraphs on $2k$ vertices satisfies

$$|H|\leq\binom{n-1}{k-1}+\left\lfloor\frac{n-1}{k}\right\rfloor.$$

Moreover, equality holds if and only if $H$ is a full $k$-star with center $v$ together with a maximal matching omitting $v$. This is motivated by the fact that the proof of the paper's main theorem uses only the absence of such $2$-regular subgraphs, apart from two auxiliary results; its resolution is not supplied in the given text.