Hong's multiplicative-function conjecture for odd GCD-closed sets

Let $S=\{x_1,\ldots,x_n\}$ be an odd-GCD-closed set, and let $f$ be a completely multiplicative function, meaning $f(ab)=f(a)f(b)$ for positive integers $a,b$. If $f$ is strictly monotone, then form the matrix $[f[x_i,x_j]]$ with $(i,j)$-entry $f([x_i,x_j])$. Hong's conjecture. The matrix $[f[x_i,x_j]]$ is nonsingular. The source says that this conjecture is false, using the function $N^\alpha$ and the paper's counterexamples.