The characterization conjecture for $F$-factors in $k$-uniform hypergraphs

Let $k\ge 4$ and let $F$ be a $k$-graph. Write $\textbf{FACTOR}_k$ for the class of $k$-graphs satisfying the $F$-factor property, let \pi_{% \begin{tikzpicture}[inner sep = 0.7pt, scale=0.11]% \node (1) at (0,-2) [aNode]{}; \node (3) at (1.5,-2) [aNode]{}; \node (2) at (0.75,-1) [aNode]{}; \end{tikzpicture}% }(F) denote the corresponding covering threshold parameter, and let $N_F(v^*)$ be the neighborhood of a vertex $v^*$. For a partition $\mathcal{P}$ of $V(F)$, let $\mathbf{i}_{\mathcal{P}}(e)$ denote the index of an edge $e$ with respect to that partition.

The characterization conjecture. The $k$-graph $F$ belongs to $\textbf{FACTOR}_k$ if and only if it satisfies:

1. \pi_{% \begin{tikzpicture}[inner sep = 0.7pt, scale=0.11]% \node (1) at (0,-2) [aNode]{}; \node (3) at (1.5,-2) [aNode]{}; \node (2) at (0.75,-1) [aNode]{}; \end{tikzpicture}% }(F)=0;
2. there exists a vertex $v^*$ and a partition

$$\mathcal{P}=\{X_1,X_2,\dots,X_{k-1},\{v^*\}\}$$

of $V(F)$ such that

$$N_F(v^*)\subseteq X_1\times\cdots\times X_{k-1}$$

and, for every two edges $e,e'\in E(F)$, if $|e\cap e'|\ge 2$, then $\mathbf{i}_{\mathcal{P}}(e)=\mathbf{i}_{\mathcal{P}}(e')$.

The paper completely resolves the corresponding problem for $k=3$; for $k\ge 4$, the stated conditions are proposed as a characterization, so the equivalence remains open.