Exponential-growth conjecture for Möbius values of the permutations $\kappa_n$

For each positive integer $n$, let $\kappa_n\in {\mathcal S}_{4n}$ be the permutation

$$\kappa_n=n+1,n+3,\dots,3n-1,1,3n+1,2,3n+2,\dots,n,4n,n+2,n+4,\dots,3n.$$

It is $321$-free and can be split into four quadrants, each an increasing subsequence of length $n$. Exponential-growth conjecture. The absolute value of its Möbius function satisfies

$$|\mu(1,\kappa_n)|=\exp(\Theta(n)).$$

Here “exponential in $n$” is recorded as exponential growth; the paper presents this as a conjecture motivated by computational experiments, and no resolution is supplied.