In general, in this setting, yes. The English (and thereby American) legal traditions assume a right "to every man's evidence" and depositions are taken under o

In general, in this setting, yes. The English (and thereby American) legal traditions assume a right "to every man's evidence" and depositions are taken under oath. An actual perjury charge is extremely unlikely, but the judge the case is assigned to could hold a witness in contempt for persistently refusing to play ball. In reality, though, what the deposing side needs is to know what facts the witness will testify to, or alternatively, to be able to show that the witness can't be trusted. So if they can prove that the witness lied, it's likely sufficient for their purposes to just show that at trial and thereby discredit the witness.