No it doesn't?
12 psig at 20C is 26.7psia at 293.15K
5C is 278.15K
The absolute pressure will reduce proportionally to the absolute temperature. (278.15/293.15) * 26.7 = 25.3 psia = 10.63 psig
In reality you'll see it read a tiny bit lower than that after cold crash - the increased solubility of CO2 in cold beer will see some of it enter solution, and there's a tiny bit of volume change attributable to the contraction of the liquid as it cools. All the same, you cold crash a beer at 12 psig you're going to be seeing 9+ PSI, not "about 1PSI".
All the charts I've seen everywhere don't reflect on that calculation but it makes perfect sense.