Hong Ri Wu, 56, of San Francisco, allegedly shot Feng Ping Ou, a 30-year-old woman, and Qiong Han Chu, a 30-year-old man, inside the souvenir and luggage shop where they worked at 269 Jefferson St. at about 8:20 p.m. on Jan. 30, police said.
Wu pleaded not guilty on Feb. 8 to two counts of murder, with special allegations of firearm use and multiple murders.
Wu's attorney, Deputy Public Defender Kleigh Hathaway, said in San Francisco Superior Court Thursday that she had doubts about his ability to participate in his own defense, according to Tamara Barak Aparton, spokeswoman for the public defender's office.
Hathaway had said outside of court following Wu's not guilty plea in February that he was being held under observation in county jail because authorities believed he was a danger to himself.
A hearing has been scheduled for June 13 for a judge to consider the competency issue, prosecutors said.
Wu is accused of walking into the victims' store and shooting them. He worked at a nearby competing store and knew Chu and Ou, police said. The victims, both San Francisco residents, died inside the store.
A gun believed to be the murder weapon was recovered at the scene.
Investigators believe the shooting stemmed from a rivalry between the two stores, which sold similar items.