You have already received the answer. I just wanted to add some comments.
If you ever have some extra time, check your books and the Web for something called “fused participles.”
Your friend has an excellent point.
a. Even the experts do not agree.
b. There are many writers who would prefer “their.”
Here are some comments from two of my books:
a. “I had a glimpse of him dodging this way and that.”
b. “I had a glimpse of his dodging this way and that.”
Two experts say: "Both are equally good. But the first emphasizes the person concerned; the second emphasizes the “dodging.”
– Pence and Emery, A GRAMMAR OF PRESENT-DAY ENGLISH.
c. “Now when [he] rides the bus to [school]. he doesn’t worry about the older kids soaking him with water guns.”
d. “Now when [he] rides the bus to [school], he doesn’t worry about the older kids’ soaking him with water guns.”
The expert says that “c” seems to be elliptical [short] for “the older kids who are soaking him”; “d” refers to “their soaking him” The expert says: " The question is what he’s not worrying about: the kids or the soaking."
–Bryan A. Garner, A DICTIONARY OF MODERN AMERICAN USAGE.
Regarding your sentence, perhaps now you can decide which word (“them” or “their”) is more appropriate.
James, As usual, you gave excellent details for the learners to dig it deeper to the problems. I already checked it. Normally, native speakers use them interchangeably. In some cases, If we do not understand, It might end up like
“smelling their moms frying chicken.” - Focus on smelling mothers rather than the frying chicken itself
Surely in the first sentence the emphasis is on them as individuals not knowing and the second sentence stresses their lack of knowledge. And while I’m on about it, ‘their knowing’ has to be possessive adjective.