"won't go through" is correct and can be used in written English.
For example, "I'm sorry, but your application won't go through this time."
"So I invented rules, such as you won't go through two chapters without meeting a real human character.
The young corals in particular studiously avoided the noxious seaweed: "There's a blanket of stink on the bottom over there and they just won't go through," Hay says.
It won't go through a tank.
"I won't go through that again without one," she said.
"I won't go through that damn thing again.
If it's too tough, it won't go through the Lords.
"At no stage has it looked like we won't go through.
"The city has made the process adversarial so people won't go through it," he said.
"I'm happy to think people who get MDR-TB today won't go through my experience.
Updated at 6.45pm BST 6.41pm BST Does anyone really think that Spain won't go through automatically?
"My writing gets teensier and teensier so I won't go through the books too fast".
