(CNN) Mystery surrounds the case of two Saudi Arabian sisters who were found dead along the banks of the Hudson River in New York City last week.

The New York Police Department held a news conference on Friday and shed more light on the case, but many questions remain. The case remains under investigation, Chief of Detectives Dermot Shea said.

Here's what we know so far:

Who were the sisters?

Rotana Farea, 23, and Tala Farea, 16, were Saudi Arabian citizens who had been living in the United States for two to three years, Shea said Friday.

They were students who were accompanying their brother in Washington, according to the Royal Consulate General of Saudia Arabia.

Rotana was formerly a student enrolled at George Mason University, according to school spokesman Michael Sandler. He told CNN that Rotana Farea was enrolled from spring of 2016 until the spring of 2018.

They were put in a shelter-like facility in Fairfax, Virginia, because of abuse allegations that came up in their home jurisdiction. Shea did not elaborate on further details and reports of abuse, which he said are not corroborated at this time.

The sisters were last seen together in Fairfax on August 24.

Tala Farea, 16, and Rotana Farea, 23.

What about their family?

The sisters were last seen by their family at their Virginia residence on November 30, 2017, authorities say.

The family last had contact with the girls in December of 2017 and had no additional contact between then and August 2018, Shea said.

Their mother apparently lives in the United States, because Saudi officials told CNN that US officials had asked her to leave the country, reportedly due to her immigration status.

How did they die?

Before the bodies were ID'd, police released sketches of the siblings.

On the afternoon of October 24, the bodies of two young women were found on the bank of the Hudson River in New York, near 68th and Riverside in an area where "it's such a place where you would essentially walk right into the water," Shea said.

The wore similar black leggings and fur-trimmed jackets. Shea said they were tied together at the waist with duct tape in a way that was "keeping them together" but wasn't meant to bind them.

The bodies were identified as the sisters on October 27.

Police have not yet determined the cause of death.

One law enforcement official, who spoke to CNN on the condition of anonymity, told CNN that water was found in the sisters' lungs, leading investigators to believe they were alive when they went in the water.

The official told CNN investigators believe the deaths were suicide.

The bodies showed no signs of trauma, according to police. On Friday Shea dismissed reports that they had jumped from the George Washington bridge, saying, "We have absolutely zero information to lead us to that."

"Is it possible? I can tell you that there were no, say, marks on the body, indicative of a fall from a height."

Ultimately, investigators have not uncovered evidence to suggest they were the victims of foul play.

"At this point in time we have no credible information that a crime took place in New York City," Shea said, "but it is still under investigation."

How did they end up in New York?

By tracing credit card usage, detectives determined they first went to Washington, DC, then Philadelphia and finally arrived in New York City on September 1, according to an NYPD spokesman.

They took multiple modes of transportation, including Uber, authorities said.

What did the sisters do before their deaths?

The sisters spent their final weeks staying at high-end hotels in New York including the Hilton and Hyatt, Shea said.

They also went shopping and ordered food -- always two meals, he said.

The expenses were put on a credit card that was eventually maxed out.

Police have seen video from as recently as last week that showed the sisters "apparently in good health," Shea said.

Detectives are also combing midtown Manhattan for surveillance footage, eye witness accounts, anything that can help them piece together what happened.

"We're looking at a two-month gap" between their disappearance from Virginia and the finding of the bodies, Shea said.

Did anyone see them just before they died?

Police said a "credible" eyewitness came forward on October 31 to share a story with detectives that was "haunting him."

The witness said he was exercising at about 7 a.m. the day the Farea sisters' bodies were discovered.

He saw two girls sitting in Riverside Park at a playground a "very short distance" from the Hudson River, police said. The girls were sitting about 30 feet apart with their heads in their hands, and were making loud noises the witness described as praying.

Was this the first time they disappeared?

No. According to the Fairfax County police, the sisters were reported missing for one day in December 2017.

Officer Tara Gerhard, a spokeswoman for the department, said the sisters were "referred to services," but she could not specify what type of services, nor offer additional details, citing privacy laws.

Were the girls seeking asylum?

Sources in the investigation have told detectives the sisters had said they would rather inflict harm on themselves or kill themselves rather than return to Saudi Arabia.

A Saudi official told CNN the girls' mother's immigration status had expired and that US officials had asked her to leave the country. But the Saudis asked the Immigration and Customs Enforcement to give their mother an extension and called their mother to let her know, the official said.

Saudi officials are still looking into whether Rotana and Tala applied for asylum, the official added. The official also refuted reports the family was told it must return to Saudi Arabia because the girls had applied for asylum.

The New York Times reported the Saudi Embassy in Washington had called Rotana and Tala's mother to inform her that her daughters had applied for asylum in the United States.

Department of Homeland Security officials declined to comment and referred questions to the NYPD as the lead on the case.