Let $G$ be a simple graph with vertex set $V$, such that for any two vertices $u,v\in V$, we have at least $k$ edge-disjoint paths of length $2$ (i.e., formed by $2$ edges) connecting $u$ with $v$. Let $n=|V|$ be the total number of vertices of $G$.

**Question:** What is the minimum value of $k$, expressed as a function of $n$, to ensure that $G$ must be a complete graph?