Final answer:
The auditor's decision to withdraw from the engagement reflects concerns primarily about their professional judgment in the face of the client's inaction on an illegal act brought to their attention.
Step-by-step explanation:
When an auditor withdraws from an engagement due to the audit client's board of directors and audit committee's refusal to act on an illegal act, irrespective of the act's materiality, the primary concern for the auditor is related to their professional judgment.
This concern arises out of the potential compromise to the auditor's duty to report and address illegality, thereby placing the auditor's ethical standards and professional integrity at risk. This action does not directly relate to doubts about the materiality of the illegal act, the legality of itself, or the auditor's independence, but rather the auditor's adherence to professional standards which expect auditors to follow up on illegal activities they uncover regardless of the size or impact.