Conversation 2.a explicitly states: 'Would you mind closing the door? (closing: object of the verb mind)'. This indicates 'closing' functions as an object, making it a gerund.
Class 6 › English Grammar and Composition › Chapter 5: The -ing form of Verb: Gerund & Participle › Topic: Verbs followed by -ing form (liking/disliking, mind, would you mind)
Aligned to the NCTB national curriculum.