The Chet Conjecture for entry-wise nonnegative Chet matrices

Let $C_n$ denote the $n\times n$ Chet matrix, and let $C=\{n:C_n\text{ is entry-wise nonnegative}\}$. Chet Conjecture. One has

$$|C|=\infty.$$

More strongly,

$$C=\mathbb{N}.$$

The paper presents this as its main conjecture. Exact computations verify nonnegativity through $n=21$, and numerical simulations support it through $n=500$, but no proof or disproof is given.