Multiplication table for orders of $(a,b)$-Fibonacci sequences with $b=-1$

Let $(a,b)$-Fibonacci sequences be defined by $\mathcal{U}_0=0$, $\mathcal{U}_1=1$, and $\mathcal{U}_n=a\mathcal{U}_{n-1}+b\mathcal{U}_{n-2}$. Let $\omega_{(a,-1)}(m)$ be the number of zeros modulo $m$ in one period. Negative-$b$ multiplication conjecture. The multiplication table for $\omega_{(a,-1)}(\operatorname{lcm}[m,n])$ is

$$\begin{array}{c|cc} \omega_{(a,-1)}(n)\backslash\omega_{(a,-1)}(m)&1&2\\\\ \hline 1&1&2\\\\ 2&2&2 \end{array}$$

Thus the order of the least common multiple is determined by the displayed table. The claim is a proposed extension of the multiplication law for $K$-Fibonacci sequences to the case $b=-1$; the source gives no resolution.