Tire and screw

Under that obnoxious title, I have a dumb question, but here I am. I noticed last evening that there is a screw in one of my tires. Towards the edge of the tire, but not on the flank.

No TPMS warning. No clue for how long this screw has been in.

I went to a tire place. They said they can’t patch as it’s too close to the edge of the tire. They sprayed some soapy water, no bubbles around the screw.

They checked the pressure, I was at 38 PSI so no major loss.

The 4 tires are a bit more than 6 months old/6 or 7k miles on them.

Here comes the stupid question: Do I keep the screw in and monitor the pressure regularly (not based on the useless TPMS)?

Actually the question is do I have to change the 4 tires or just the 2 on the axle impacted? Since it’s a Quattro, I’d rather ask.