'Their' technically could be used, if 'he' is talking about two or more other people's principles, and not his own. Duo usually likes the possessive adjective to agree with the subject 'El' and the count of the object. I hope that you reported it, because in reality he could possibly be referring to two or more people. Most of the time, Duo will put 'de ellos/as' when there are two or more people.
Not everyone was in a lesson about directions; I was in "vocabulary."
Duo accepted my use of "beginnings," & dinged me for using "their." I thought it would be quite natural if a man who owned a successful restaurant, for example, was telling someone about how he & his wife started out in their married life running a hot dog cart, or a small catering service, because then he would speak of the way THEY began (their beginnings in business), requiring the plural possessive pronoun, but if he was speaking only of HIS start in life, it would have been su. Given "sus," I think "their" should be accepted as an alternate correct answer. I will report it; if enough people do, they may change the accepted answers or change the sentence by adding "*los ellos" for clarity, perhaps.
The fact that it could also mean "principles" complicates the responses a lot, if a person has to make the possessive pronoun plural to go with the plural "principles"! Then you really would need the added context of adding a él o los ellos to apply clarity to "sus." If my answer is not reasonable, I would appreciate an explaination as to why not, rather than just dinging me with a down-vote, please, since Duo-owl already did! ;-)
Maybe, but you could also say with no other context we have to go with the most likely and that would be he is talking about "his" own "principles". I do not see a native Spanish speaker holding all 6 (his, her, one's, their, its, your) possibilities in their head at the same time when they hear this. That would be utter madness.
More sentences like this ("The more he talked of his honor, the faster we counted our spoons.") could make for some VERY interesting discussions about the "best" way to translate. In any language, there are often several ways to say the same thing, but first the translator has to figure out what's being said!