DNA test results were not released during the closed-door session in the Bahamian Supreme Court. And the judge scheduled another hearing for next week.
Neither Birkhead nor Howard K. Stern attended yesterday's court hearing. Their lawyers are bound by Bahamian legal rules and aren't allowed to talk about specifics of the case.
Sources tell "Entertainment Tonight" that Smith's estranged mother, Virgie Arthur, has filed papers in the Bahamas in hopes of becoming the infant's guardian ad litem. A hearing is set for April 13.
The Supreme Court completed a scheduled hearing Tuesday but the public might not find out the result of that hearing until next Tuesday, ET reports.
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"I think most people believe Larry Birkhead has more swimmers in the DNA pool from late 2005, so he's on the odds-on favorite. Howard K. Stern is still alive as potential father. He will not be ruled out today unless it's Larry Birkhead on that DNA test," Court TV correspondent Michel Bryant told The Early Show co-anchor Julie Chen on Tuesday.
Stern, the lawyer-turned-partner of Smith, dropped his bid Monday to throw out DNA test results for the late Playboy Playmate's infant daughter when it appeared a Bahamas court was going to reject his appeal.
An attorney for Stern withdrew the challenge in the Court of Appeal in the face of skeptical questioning by a three-judge panel.
All three judges said that Stern had filed his challenge too late and should have raised his objections before the Supreme Court ordered DNA testing in the paternity challenge filed by Birkhead.
Justice Emmanuel E. Osadebay noted that Stern himself had agreed to DNA testing and waited until a week after the sample was taken to file the challenge.
"Not only was his attorney embarrassed and had to withdraw what was there appellate application, but on the way out, the court said, 'you wasted everybody's time here,'" Bryant said.
Stern is listed on the birth certificate as the father of Dannielynn, who was born in the Bahamas in September.
Bryant said that if Birkhead is found to be the biological father, he won't automatically get custody of Dannielynn.
"If it is Larry Birkhead, it's likely social services will get involve involved. The custody battle will then heat up," Bryant said. "You've got Virgie Arthur, the maternal grandmother who says she wants custody, and Larry Birkhead will be fighting for custody as well. It wouldn't be resolved completely but would certainly be focused."
The baby, whose full name is Dannielynn Hope Marshall Stern, could inherit millions from the estate of Smith's late husband, J. Howard Marshall II. Smith had been fighting the Texas oil tycoon's family over his estimated $500 million fortune since his death in 1995.