Does the gain conversion from Infernal Cry empowered attacks count as Inherent to the skill?

Infernal Cry, says

"Empowered Attacks Gain (30—49)% of damage as extra Fire damage"

Gain Says

Damage Gain occurs in the same two step process as Damage Conversion. Damage over time cannot benefit from damage Gain.

Damage conversion says

Conversion is a two step process. Conversion inherent to Skills occurs first, then Conversion from all other sources. Damage over time cannot be converted.


Essentially I am wondering if Xoph's Pyre synergizes with infernal cry empowered attacks.

Or getting a clearer definition of what "inherent" counts as.

The gain conversion of these empowered attacks is inherent to the warcry itself, but does it no longer count as inherent due to converting the damage of other attacks instead of the damage of the skill itself? Or does inherent just mean it is damage gain provided by the base functionality of a skill itself rather than an item or support gem?