An Inheritance Issue
The answer to whether Aldrich would have a showdown with Solomon Solar would all come down to what V could extract from Desmond's mind.
Desmond was potentially the biggest informant for the Italian Prong of the Trident, so he had valuable information from the top down about the prong's operations and secrets.
Incidentally, it was the Italian Prong that seemed to have the closest ties with Solomon Solar.
Of the three prongs, the Italian one could be considered the 'business' side, dealing less with combat and more with navigating deals with other organizations, managing trade routes for drugs and illegal goods setting up illegal experimentation centers.
The Japanese prong was the muscle. They were almost solely focused on cultivating the strongest fighters, creating an in-house supply of top notch, loyal talent to field to make sure no other organization thought about taking a fight with the Trident.