A conflict is representing someone against someone you have represented in the past, or are currently representing.
Perhaps what you meant was exactly what the law firm stated... that the partner was not authorized by the firm to file the lawsuit....which has nothing to do with whether a conflict (defined above) exists between the firm, the Pillow Guy and Dominion.
Here is an ABA primer for you:
https://www.americanbar.org/groups/professional_responsibility/publications/model_rules_of_professional_conduct/rule_1_7_conflict_of_interest_current_clients/
https://www.americanbar.org/groups/professional_responsibility/publications/model_rules_of_professional_conduct/rule_1_8_current_clients_specific_rules/