Asymptotic existence conjecture for $(a,b)$-necklaces

Let $a$ and $b$ be positive coprime integers. An $(a,b)$-necklace is the combinatorial object considered in the paper, with length denoted by $n$. Asymptotic existence conjecture. There is a positive integer $n_{a,b}$ such that an $(a,b)$-necklace of every length $n \geq n_{a,b}$ exists when $ab$ is even, and an $(a,b)$-necklace of every even length $n \geq n_{a,b}$ exists when $ab$ is odd. The conjecture removes the sufficient restriction $2a\leq b$ from the preceding existence theorem; numerical computations support it, but the general assertion remains open.