Translation:You think they would open the door for us?
In French, it's structured as a phrase, but implied as a question.
The English translation is the same way.
I realize it may come across as a little weird, because earlier Duolingo lessons would have the French only implied as a question, but the English actually structured as a question. I never liked that.
So wherever possible, I try my best to use "est-ce que" or the subject-verb inversion when I'm translating an actual structured question written in English into French, as opposed to just simply writing a French phrase, and putting a question mark at the end, like Duo would rather have me do 80-90% of the time.
Here "would" is used as a conditional verb (since it is not in past tense, and therefore not the past tense of "will") but this is not an imagined situation, but could happen.
So "will" should/could be used:
"(Do) you believe/think they will open the door for us?".