"Toxic Growth" skill doesn't apply Wither when supported by "Withering Touch" support

The title.

Is you Toxic growth doing chaos damage to triggering withering touch?

You can check it in the arrow button showing the skill details, including damage type done.

Toxic growth is a chaos skill but by default the explosion are physical damage if your bow don't have any chaos damage mod or gem support "chaos infusion". Exactly like rain of arrows.