A property on reinforcing edge-disjoint spanning hypertrees in uniform hypergraphs

Suppose that $H$ is a simple uniform hypergraph satisfying $\left|E\left(H\right)\right|=k(\left|V\left(H\right)\right|?1)$. A $k$-partition $\pi =(X_1,X_2,\dots ,X_k)$ of $E\left(H\right)$ such that $\left|X_i\right|=\left|V\left(H\right)\right|?1$ for $1\le i\le k$ is a uniform $k$-partition. Let $P_k\left(H\right)$ be the collection of all uniform $k$-partitions of $E\left(H\right)$ and define $\epsilon \left(\pi \right)={\sum }_{i=1}^{k}c\left(H\left(X_i\right)\right)?k$, where $c\left(H\right)$ denotes the number of maximal partition-connected sub-hypergraphs of $H$. Let $\epsilon \left(H\right)={min}_{\pi \in P_k\left(H\right)}\epsilon \left(\pi \right)$. Then $\epsilon \left(H\right)\ge 0$ with equality holds if and only if $H$ is a union of $k$ edge-disjoint spanning hypertrees. The parameter $\epsilon \left(H\right)$ is used to measure how close $H$ is being from a union of $k$ edge-disjoint spanning hypertrees.We prove that if $H$ is a simple uniform hypergraph with $\left|E\left(H\right)\right|=k(\left|V\left(H\right)\right|?1)$ and $\epsilon \left(H\right)>0$, then there exist $e\in E\left(H\right)$ and $e^{\prime }\in E\left(H^c\right)$ such that $\epsilon (H?e+e^{\prime })<\epsilon \left(H\right)$. This generalizes a former result, which settles a conjecture of Payan. The result iteratively defines a finite $\epsilon$-decreasing sequence of uniform hypergraphs $H_0,H_1,H_2,\dots ,H_m$ such that $H_0=H$, $H_m$ is the union of $k$ edge-disjoint spanning hypertrees, and such that two consecutive hypergraphs in the sequence differ by exactly one hyperedge.