SAN JOSE — The woman in the bright pink tank top might have known too much.

San Jose police have yet to identify her, but in a captivating news conference Tuesday, they suggested that if they hadn’t gunned down a chief suspect in a homicide at an office building last week — shooting him as he fled in his red sedan with the woman in the passenger seat — she might have been next.

The dramatic case remains riddled with questions and has left police defending themselves for killing two of three men caught on the office building security tape hunting down ex-con Christopher Wrenn, who is seen on the tape in a beanie cowering with his hands up before he lay dying in a pool of blood.

One of the suspects was shot by police in front of a laundromat in South San Jose off Monterey Highway on Sunday night; the second, Richard Jacquez, was shot by police Monday evening as he ran toward his cousin’s house in a cul-de-sac a half-mile away. The third suspect, considered armed and dangerous, remains on the loose and police are hoping members of the public call in with leads.

The woman in the pink tank top, however, may have known details of the midnight homicide last Thursday at a half-empty office building on Lundy Avenue in a tech area of North San Jose.

“There was probable concern about what information this female had about the homicide and the suspect’s concern about her divulging that information and putting him in jeopardy of being apprehended,” police spokesman Sgt. Enrique Garcia said at Tuesday’s news conference. He declined to name the suspect, but Jacquez’s grown daughter and cousin, interviewed in the cul-de-sac Tuesday morning, said it was him.

Relatives planned a vigil Tuesday evening near the shooting scene on Kirkhaven Court. They say police acted recklessly.

“Whatever he did, they didn’t have the right to chase him down like a dog,” said Jacquez’s cousin, Danny Aguirre, 23, who said several family members live in the mustard-colored house that Jacquez was running toward on Monday night. He died on the front stoop.

Police say the suspect wasn’t armed when an officer shot him in the back, nor was he reaching for his waistband, as they had earlier suggested. But they say that, based on seeing him carrying a weapon in the office building video, they had no choice but to presume he was armed and an ongoing threat.

“The officer felt he had to make a decision,” Garcia said. “If he allowed this person to break into or go into this house, we have a barricaded, hostage situation, and we know this person is suspected of having murdered someone else.”

Neighbors on Kirkhaven Court recounted the terrifying episode that went down at dinnertime Monday. Drawn to the front windows when they heard police sirens, they then heard the crash of a red sedan as it careened into a neighbor’s parked car. Police who were tailing him called out to the man as he ran toward his cousin’s house, neighbor Anita Rosas said. Neighbors then heard the “bang, bang, bang” of gunshots.

By this time, the woman who had been a passenger in the car was standing next to it, screaming.

“She was crying and waving her arms. She was all upset,” Rosas said, but the woman managed to make a call on her cellphone. Rosas found out Tuesday morning, when Jacquez’s daughter knocked on the door, that the woman had called her in the midst of the chaos to say “come quick,” that her father had been shot by police.

The daughter told Rosas that the woman was a friend or girlfriend of her father’s. The daughter also asked whether Rosas had a home security camera that might have caught the shooting on video. Rosas didn’t.

In an interview Tuesday, the daughter, who didn’t want to be named, said that her father was a loving person and she couldn’t believe he would have killed someone. She said he was a gentle grandfather to her 1-year-old daughter.

“He was willing to do anything for anyone,” the daughter said. “I trusted him with my life and her life.”

About 100 people gathered Tuesday night on the corner of Stoneyhaven Ave and Kirkhaven Court to mourn the loss of the man they knew as father, brother and friend — not the man wanted in connection with a homicide. Jacquez, they said, was a San Francisco Giants and San Francisco 49ers fan. He loved his family. They knew he was flawed, but they adored him anyways.

“My uncle was a great person,” Jacquez’s niece Miranda Montiel said at the vigil. “When he was shot, he was unarmed.

Shooting him wasn’t OK. They didn’t have to kill him. We were all close to him. Yeah he may have committed crimes, but he was a goodperson.”

Police say, however, that the security video from the office building shows that Jacquez had been armed with a Tec-9 assault weapon. That same weapon, they say, was recovered overnight Tuesday when officers served a search warrant at an undisclosed San Jose home. Court records show Jacquez had been convicted in 2008 for felony robbery.

The second suspect, killed Sunday, was identified by police Tuesday as Matthew Castillo, 29, of San Jose. A gun was found on the scene at the laundromat and an eyewitness there said he saw Castillo holding the weapon before police shot him.

Little has been disclosed about the third suspect, seen on the office building security camera as heavyset, wearing all black and his face covered in a bandanna.

“We really could use some leads from the public regarding identifying that third individual, who is considered armed and dangerous,” Garcia said. “We’re hoping that this person realizes that it’s in his best interests to surrender, through family members or friends or come into the police department and talk to us, and let’s try to resolve this peacefully.”

Police said anyone with information about the case can contact homicide detectives at 408-277-5283 or leave a tip with Silicon Valley Crime Stoppers at 408-947-STOP or svcrimestoppers.org.

Staff writer Katie Nelson contributed to this report.