Integrality conjecture for ordered multi-degree contributions

Let $(d_0,d_1,d_2)$ be an ordered multi-degree, and let $\operatorname{C}_{\operatorname{ord}}(d_0,d_1,d_2)$ denote its ordered multi-degree contribution. Write $\operatorname{gcd}(d_0,d_1,d_2)$ for the greatest common divisor of the three degrees.

Integrality conjecture. The ordered multi-degree contributions satisfy

$$\operatorname{gcd}(d_0,d_1,d_2)^2\cdot\operatorname{C}_{\operatorname{ord}}(d_0,d_1,d_2)\in\mathbb{Z}_{\geq 0}.$$

This predicts a divisibility and nonnegativity property for all ordered multi-degree contributions. The supplied text gives no resolution or further evidence sufficient to determine whether it is known, so it remains open.