Switch design for shorting the power control pins?
You have done a good job at defining the task, but not the interface logic levels and outputs. Please update your specs

for more test results in case something is damaged on your interface measure Iin, as Vin is varied from 14.2 to 0V with a variable current limited supply to get a better handle on the problem performed with a triangle ramp in &lt;&lt; 5 seconds. Use a transistor to control the current.  (Since the user manual is vague)  Also, confirm the pressure switch does not leak uA.

50uA if fails for Return to High seems too sensitive and 143 mA seems too high for a logic low Start switch.


From the OPN-USB Manual for J8 p5/12
Starting and Stopping with battery power….

If everything correctly configured pressing shortly the J8 button will turn on the output,
and your UPS will be started, energy will be flowing from the battery to the output.

Long pressing (5 sec) J8 button will initiate shut down of the UPS. A pulse is sent 
to the motherboard than the UPS waits a predetermined UPS_HARDOFF_TOUT (60s by
default) before the output voltage is turned off.

i.e. Start is ---_ ---- and Stop is ----______------- ,
where --- = open switch and  ___ is closed switch yet pressure sense is normally open with dial in-tight and switch closes to 0V to exceed long duration &gt;5s to Stop OpenUSB power.
Your problem is commonly defined in datasheets as\$I_{CBO}\$ which is the collector-base cutoff current at rated Vce max voltage with Ie=0.
This is due to the “Early Effect” as a reverse diode leakage across the CB junction which has a load line dependent on Vce.
Although the 2N3906 is rated at 50uA it is only rated at Vce=-40V so choosing a higher voltage say &gt;100 V , you will find components with lower max Icbo leakage , which also rises above 25’C much lower.
For example if it says 250nA at -300V, this is equivalent to &gt;1GOhm but again will rise in temp and this may be nominal.  This also makes it sensitive to stray EMI during this state with electromagnetic noise coupling so shielded twisted pair may be needed if there is any EMI nearby during this state.
You can lower the input impedance by using a Common Base mode driving the emitter as a switch with some bias R divider on the base from 12V to Gnd such that using hFE = 10% of linear hFE max would get reasonable Vce(sat) to drive 125mA with the switch.
This should solve the leakage problem.
added
I suspect your actual circuit looks more like below with ground switched power activation but unfortunately with 50uA (rated? or measured?) leakage to ground somewhere.
The 2N3906 is one such path.

Not a solution, just maybe an answer to define the problem.