Excuse my jumping in here, but I'm having the exact same issue. Changed CTS, GP still dont stay on long enough, unless CTS is unplugged.
I followed the troubleshooting and schematic above, and got the following:
CTS plug:
pin 1- very low voltage
pin 3- 5V
T14a:
open on pin 9 to gnd
continuity on both wires to the CTS plug.
9V on pin 4 on the female side of T14a....
So, is that a bad ground at T14a pin 9? If so, instead of hunting or ripping out the dash like detailed above, why not just spice in a gnd near T14a off of the brown pin 9 wire?
It got dark before I got to figure out the reason that I had 9v at t14a but nothing at the cts, despite showing continuity through the purple wire. bad connection at t14a maybe? I didnt notice anything visually wrong there.