Binary System not benefiting from cold damage

I did some testing with Binary System (Black Hole), the passive state it spawn a hole with Cold Damage (in addition to the fire dmg BH), but i did some test, with two rings (one with cold and the other with fire damage). In theory if one of the holes deal cold, the ring with cold suppose to increase the damage but only the ring with Fire increase the damage
I’m unsure if that’s intentional, a bug or an oversight related to the description