pronouns - They stayed that way: he/him/his playing his guitar, she/her singing her songs

I've found other threads (see example links below), but I still can't sort this one out on my own. Are these -ings gerunds or participles? And therefore should they be preceded by accusative, possessive, or nominative pronouns?

They stayed that way: he/him/his playing his guitar, she/her singing her songs.

Helpful Example Threads:

When is a gerund supposed to be preceded by a possessive adjective/determiner?)

What's the difference between a gerund and a participle?