Triumvirate and Hateforge Rage Cost Interaction

When using a Triumvirate Authority ring with the modifier "Socketed Vaal skills require 30% less souls per use" together with the Hateforge gloves, the Rage cost seems to be unaffected by the Triumvirate modifier.

I would have assumed that the Soul cost modifiers is applied to the Vaal skill and afterwards it's transferred to a Rage cost via the gloves. I tested this on the example of Vaal Lightning Strike:

Vaal Lightning Strike: 20 Souls
Triumvirate 30% less => 14 Souls
Hateforge => 14 Rage
25% reduced rage cost (from hateforge) => 11 Rage

But it stays at 16 Rage, i.e., the "less" modifier is not applying.

Is this intended?

I uploaded the screenshots here: https://imgur.com/a/gA5weac

Best wishes!