Superlinear growth conjecture for complete pivoting

For each positive integer $n$, let $g\big[\mathbf{CP}_n(\mathbb{R})\big]$ denote the maximum growth factor for complete pivoting over real $n\times n$ matrices. Recall that $f(n)=\omega(h(n))$ means

$$\lim_{n\rightarrow\infty}\frac{f(n)}{h(n)}=\infty.$$

Superlinear growth conjecture. The maximum growth factor is superlinear:

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

The conjecture is motivated by numerical evidence for matrix sizes up to $75$ and $100$; the paper also reports a lower bound on the limiting superior ratio, but an exact asymptotic estimate remains elusive.