(CNN) —  

The UK court presiding over the case of Charlie Gard will make its decision Wednesday on where and when the terminally-ill child will die.

In a surprise move, the family’s lawyers and mother Connie Yates returned to the High Court on Tuesday after abandoning legal action the previous day over treatment for the 11-month old, who is currently on life support.

At stake is whether his ventilation tubes should be withdrawn in the hospital, in hospice, at home soon after transfer or at home after a period of days.

Charlie Gard’s parents want to bring their son home from hospital to die. But the hospital argued Tuesday that there were no doctors able to oversee Charlie’s death at home. The court has given the family until noon tomorrow to find a medical practitioner prepared to take up the case.

“I will make my final decision tomorrow unless there is something that comes up tomorrow,” said Justice Francis, who is presiding over the case.

The court heard Tuesday how London’s Great Ormond Street Hospital (GOSH), which is caring for Charlie, had concerns about getting the right medical equipment into Charlie’s home and had suggested a hospice and a mediator. Lawyers for Charlie’s parents said they were willing to pay all costs and had other suitable family properties if needed.

“I am sensing that timing is a lot of the problem here,” said the judge. “It now seems to hinge partly on how long parents want to keep Charlie alive at home.”

Hospital: US doctor offered false hope to parents

On Monday, Chris Gard and Yates gave up their fight to take Charlie to the US to be treated by Dr. Michio Hirano, a professor of neurology at the Columbia University Medical Center in New York.

GOSH accused Hirano of bringing false hope to the baby’s parents. In a statement on Monday, the hospital also expressed surprise that the doctor had a financial interest in the treatment he was offering.

Hirano told the court in mid-July that a treatment known as nucleoside bypass therapy (NBT) had a small chance of bringing about significant improvement in Charlie’s form of mitochondrial DNA depletion syndrome.

“When the hospital was informed that the Professor had new laboratory findings causing him to believe NBT would be more beneficial to Charlie than he had previously opined, GOSH’s hope for Charlie and his parents was that that optimism would be confirmed,” said the GOSH statement.

“It was, therefore, with increasing surprise and disappointment that the hospital listened to the Professor’s fresh evidence to the Court.”

GOSH also accused Hirano of not having read Charlie’s medical or court records before putting forward a treatment.

Hirano’s UK lawyer did not immediately respond to CNN’s request for comment.

This story has been updated to reflect the latest developments.