The Motzkin product conjecture for entangled threads

Let an even thread be a thread of even type and an odd thread be a thread of odd type. For an even thread $\alpha$ and an odd thread $\beta$, consider the threads entangled with each of them. Motzkin product conjecture. For any even thread $\alpha$ and odd thread $\beta$, the number of odd threads entangled with $\alpha$, respectively the number of even threads entangled with $\beta$, is a product of Motzkin numbers. This conjecture predicts a uniform product formula extending the product formulas established for layered even and odd threads; proving it remains open.