Hong's odd GCD-closed-set conjecture for real powers

Let $S=\{x_1,\ldots,x_n\}$ be an odd-GCD-closed set, meaning that $S$ consists of odd positive integers and is closed under greatest common divisors. For a real exponent $\alpha$, let $[[x_i,x_j]^\alpha]$ be the matrix whose $(i,j)$-entry is $[x_i,x_j]^\alpha$. Hong's conjecture. If $\alpha\neq0$, then the matrix $[[x_i,x_j]^\alpha]$ on $S$ is nonsingular. The source does not assign a resolution to this formulation in the candidate span.