The woman, 28, was heard saying that she had "blacked out and drowned her babies."
Artem Eydelman admitted to beating, shaking and squeezing his daughter until she lost consciousness.
"We cannot go and force our way in," said local police chief Dependera Pathak.
Pike County prosecutors said the custody of suspect Edward "Jake" Wagner and victim Hanna May Rhoden's daughter played a role in the slayings.
Police are now investigating the deaths of three other children he fathered.
Police have arrested four family members over the death.
The suspect told police he found his fellow serviceman in the closet of his wife's bedroom.
Ramos will be the eleventh prisoner executed by Texas so far this year.
A couple in Shreveport, Louisiana, were reported missing by their children after they gave an unknown man a ride from the mall.
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Two U.S. Marines familiar with the incident told Newsweek that Rodolfo Rivera strangled his wife Natasha Rivera after the couple had returned to their hotel room.
"He is a soulless individual who thought he was gonna outsmart everyone," said Volusia County Sheriff Mike Chitwood.
"Reports that we ordered anyone related to the Saudi sisters...to leave the US for seeking asylum are absolutely false," an embassy spokesperson said.
Yasin Aktay, the top adviser to Turkish president Recep Tayyip Erdogan, said the Saudis wanted "to leave no trace behind."
The remains of a missing 61-year-old, who had frequented the restaurant, were discovered in the kitchen.
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The man was allegedly caught posting an advert seeking a girl to facilitate his necrophilia and cannibal fantasies.
Saudi Crown Prince Mohammed bin Salman promised that "justice will prevail" after Jamal Khashoggi's death, but even the U.S. has begun to doubt his story.
The soldier's mistress allegedly knew he had killed his wife and appeared to be helping him dispose of the body.
Jared Kushner said that the administration must counter "people who are trying to deceive us" but also stay focused "on what's good for America."