"Kelly remains committed to having her children safely returned to their country," a source close to the actress tells PEOPLE

By Michele Corriston
Updated June 03, 2015 12:05 PM
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Cindy Ord/Getty; Rukhsana Hamid/Bloomberg News

Kelly Rutherford came closer than ever before to bringing her kids back to the United States in May – but days later, she hit another obstacle in her six-year custody battle.

Last week, a California judge halted a court order filed May 22 that gave the Gossip Girl star temporary sole custody over her young son and daughter to fly them to Los Angeles from Europe. The children, Rutherford and her ex-husband Daniel Giersch are due in L.A. Superior Court on June 15 to evaluate the current parenting arrangement, in which the estranged couple technically share joint custody. Hermes, 8, and Helena, 6, have lived with their dad in Monaco and France since 2012, when his U.S. visa was revoked.

Now, Rutherford, 46, must wait to see how a June 11 call between Monaco and L.A. judges pans out.

“Kelly remains committed to having her children safely returned to their country and looks forward to raising awareness for those who have found themselves in a same or similar situation through her organization, the Children’s Justice Campaign,” a source close to the actress tells PEOPLE.

According to the legal papers Giersch filed Thursday, the German businessman asked for a stay on the earlier court order because he claims Rutherford made “false claims” about him “depriving her of custodial time” while obtaining it. (Last month, she told PEOPLE Giersch would not let her see the kids after she arrived for her latest visit unless she turned in their passports.)

Giersch also asserts that Hermes and Helena can’t be “returned” to L.A. when Rutherford doesn’t live there anymore; she is based in New York. He’s asking for a hearing on whether California has jurisdiction over the children.

“Daniel continues to guard the privacy and best interests of the children,” his attorney, Fahi Takesh Hallin, tells PEOPLE in a statement. “Daniel’s only goal is promoting the children’s relationship with both parents.”

What’s at stake June 11? Nancy Chemtob, a New York family and divorce lawyer not involved in the case, says the judges will just be deciding where the next hearing should take place. Whether they proceed in California or Monaco will have no bearing on the custody arrangement or outcome, she explains.

Despite the earlier order, there is a big chance the case could be moved to Monaco even though the L.A. court previously asserted jurisdiction over the kids. According to the Hague Convention, children’s residency is where they have been living for six months – and Helena and Hermes have lived abroad for nearly three years.

“A custody case would be heard where the children have more connections, and right now, I would believe that the children would have more connections in Monaco,” Chemtob says. “It’s really what’s more convenient for the children and where the case would get a better result on the facts – whether or not the children’s teachers would be called to testify or their doctors or their friends’ parents.”

Though Chemtob says it’s possible for two countries to both have jurisdiction, she adds, “What might be a concern to the court in Monaco is that the proper forum for this case to be heard would be Monaco, because that’s where the necessary parties are, where the children are residing currently, there is no emergency, and maybe whatever [Rutherford] said last time that made the judge act, [Giersch’s team is] now saying isn’t true.”

Chemtob advises Rutherford to gather any more information supporting her claims that Giersch sought to alienate her from their children and get it to her lawyers before the June 11 conversation.

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If the judges decide to move the hearing to Monaco, Rutherford could still try to appeal – but if the case remains in California, Giersch would have to comply with the original court order and allow the children to travel to the United States before June 15, she explains.

“The judges are just trying to clarify which judge wants to or should hear the case,” she says.