Is inquiring into jury bias constitutional?

With a plaintiff-side background, I've definitely heard the clichés about greedy litigants faking injuries for the sake of a windfall, and although jurors are meant to give a fair hearing and determine facts based on evidence, bias often deprives one side or the other of due consideration.  Per SCOTUSblog, next term the Supreme Court will consider the case of Pena-Rodriguez v. Colorado and address whether parties can have both an impartial jury and a jury entitled to confidential deliberations.

Apart from the obvious tension between confidentiality and prying into an internal process to assess impartiality, there's also the problem that confidentiality could be necessary to allow impartiality to prevail over bias.  The issue in Pena-Rodriguez, for instance, arose from racist comments made by one juror, in a criminal case which resulted in a guilty verdict for a Mexican defendant.  If the racist juror believed that other jurors would tell the defense attorneys about his comments, it's likely he wouldn't have made the comments to begin with.  This would have kept his bias completely under wraps, never revealing to the court that his vote might have been based on pure prejudice, and never revealing to his fellow jurors that he had motivations besides what the evidence showed, which could undermine the influence of his vote.  Concerns that deliberations were not confidential would interfere with the ability to bring the absence of impartiality into the light.

Still, even the possibility that one's biased statements would be disclosed could render a juror less likely to make the statements to begin with, depriving his or her fellow jurors of the opportunity to account for those statements.  If bias isn't known and accounted for, no decision can ever be impartial.

And what about the lawyers' job?  From an appellate perspective, what's really galling is that the judge apparently advised defense attorneys that past jurors had expressed dislike of individuals who entered the U.S. illegally, but defense lawyers did not screen potential jurors for this possibility.  Is there any effect if the lawyers don't inquire into a specific possible bias during jury selection?  Who knows if this will play a large role in the Court's eventual decision - at any rate, no one ever lost a case from preserving issues.

Maybe this case will result in a situation similar to jury nullification, which juries can invoke but the court isn't required to instruct them on it.  At any rate, it could provide a basis for finding that plenty of parties did not receive the fair hearing they're entitled to, and even more so if the lawyers lay a foundation for that claim.