Optimality conjecture for trivariate Lissajous curve frequency triples

Let $(a,b,c)$ be a triple of strictly positive integers, and let $c_n$ be the largest entry of the frequency triple specified by the paper's equation. Assume

$$\max\{a,b,c\}<c_n.$$

Optimality conjecture. There exists an integer triple $(i,j,k)$, not all zero, with $i+j+k\leq 2n$, such that at least one of

$$ia=jb+kc,\qquad jb=ia+kc,\qquad kc=ia+jb$$

holds. Equivalently, the triples specified by are precisely those satisfying the conclusion of Theorem 1 with the minimum possible maximum frequency.

The conjecture asserts that the frequency triples in give optimal rank-1 Chebyshev lattices for the stated cubature degree. The supplied text does not indicate whether this optimality claim has been proved or disproved.