Complete-pivoting threshold conjecture for the maximum growth factor

For each positive integer $n$, let $g\big[\mathbf{CP}_n(\mathbb{R})\big]$ be the maximum growth factor for complete pivoting over real $n\times n$ matrices. Complete-pivoting threshold conjecture. The growth factor satisfies

$$g\big[\mathbf{CP}_n(\mathbb{R})\big] \ge n$$

if and only if $n>10$. The claim is based on numerical optimization: the authors report that extensive computations at $n=10$ never exceeded $9.96$, while exact maximal growth factors are known only for $n=1,2,3,4$.