Can a Judge Refuse to Conduct a Gay Marriage?

Here's one right in the Vischer wheelhouse.  This is the question that an anonymous New York judge asked the New York Judicial Ethics Committee.  In this judicial ethics opinion, the Committee largely did not answer it, though it did opine that the judge could choose to conduct only those weddings of his relatives and friends.  That would be tantamount, in the Committee's view, to refusing to conduct marriages "on a facially neutral basis" and the Rules Governing Judicial Conduct do not require a judge to conduct weddings.