I'd think "immaginare' is used to express something that one is in fact "imagining" in the sense of "has in mind" as in: "I can imagine myself living in Italy." --in which case none of the other synonyms would apply. "Supporre" on the other hand is I think used when a synonym for to "imagine" could be to "assume" or "suppose" as in: "I imagine, meaning I assume/suppose it's hot in Italy right now." While I'm not perfectly sure (whatever is?) I imagine that's the difference. (Or maybe it's I assume that's the difference).
To imagine something and to suppose aren't synonymous. "I suppose so" would be "Suppongo cosi'". Given the inclusion of the direct object "lo" I don't thing it can mean "I suppose so". You couldn't e.g. say "I suppose it." I think Duo's translation is the correct one here.
They are in some contexts - "I suppose DL's always right/I imagine DL's always right." Certainly in other contexts they're not interchangeable -- in "I like to imagine a world at peace" the verb can't be replaced by "suppose", but in some they certainly are....at least I imagine they are -- or is it 'suppose'?
You make a valid point, but these exercises shouldn't only be about translating words, rather and more importantly they should be about translating, i.e., expressing ideas in another language accurately and in a way that's natural and grammatically acceptable for that language. Translating word for word, especially with cognates that aren't always used the same way in different languages, despite their common roots, isn't what learning a language is all about - it's about the expression of ideas, not individual words, so if in this particular sentence, both 'imagine' and 'suppose' express the idea then what difference does it make which one you choose. You're correctly expressing the idea in English - isn't that what learning a language is about? Now that said about translating in general, to use "suppose' for 'supporre' here because it's in some contexts a legitimate cognate, just doesn't work: "I can suppose it" makes absolutely zero sense. You wouldn't be understood and you'd be asked to clarify what you mean. Here you must go with an alternate choice for "supporre" namely "imagine."
This is what Duo says when you use "I suppose so". In corrections, it changes as little as possible rather than giving its preferred version, hence "so" becomes "it" but "suppose" survives. Remember this when you get other stupid corrections.
Unfortunately we then lose the point of the example, which is to teach a common phrase in both languages. BTW "I suppose/imagine so" (not "it") is suppongo così, rather than di sì, which is "yes", I suppose :-)