Round 3:
Before beginning "Round 3", just a brief summary of where we are, and how we got here, so everyone is on the same page:
1) Car is 2012 328xi N52 E91, which overnight developed "Crank, NO Start", with fault codes for DME Relay Activation, and NO voltage at F37, which is fuse powered via DME Relay, that provides Main Voltage Supply to the DME Module on ALL 2008 & later E9x models. On E9x models built after 3/1/2007, that DME Relay is NOT a "Plug-in" relay, but rather soldered to the JB Fuse Panel Printed Circuit Board (PCB).
2) You tested Function of the DME Relay by applying a Chassis Ground Jumper to X11010/1, or Pin #1 of the Connector X11010, on the upper-left of the JB Fuse Panel. This caused the relay to "Click" or "Activate", providing power to F37 (and the DME), and also providing power to fuses F38 & F39 which provide power to engine components such as Injectors, Coils, and various Sensors.
3) You were then able to Start the Engine when Starter was cranked, as the DME, Coils, Injectors, and Sensors such as Crankshaft Sensor, Camshaft Sensors, etc. were NOW properly powered.
4) You tested for Continuity to Ground at the White/Gray wire from X60005/13 at the DME, testing at pin #5 of the intermediate Connector in the E-box, X6011/5. There was NO Ground signal or continuity to Ground with Ignition ON. There SHOULD have been such an "Ignition-switched Ground". You WERE able to apply a chassis ground jumper to the Red Gray wire running from X6011/5 to X11010 at the JB, and thereby Activate the DME Relay. You SHOULD do "Continuity Test" to rule out the last slim chance that there is a wire fault in the White/Gray wire between X6011/5 and X11010, and assuming NONE is found, proceed as described below.
All those tests indicate the DME Relay functions, is getting 12V+, but is NOT being activated by the Ground Signal that the DME SHOULD be sending from X60005/13 to X11010, to cause power to flow through the Relay Electromagnet, close/ "Pull-in" the relay contacts, and provide power to fuses F37, F38 & F39. So you have TWO choices:
1) Wire in a Toggle Switch to Manually Activate (and de-activate when turning engine off) the DME Relay by wiring-in a Chassis Ground Jumper completed by toggle switch, "T'd" into the Red/Gray wire at X11010/1. Although this is the quickest & easiest way to restore engine operation, it requires mounting such a toggle switch somewhere, and remembering to use it. It has the advantage of being a "One-Off Immobilizer" or Theft-deterrent, and the disadvantage of having to remember to turn it on/off, and the NON-BMW appearance.
2) Restore original function by doing one of the following:
a) Remove the DME, disconnect & open Module 5, and examine X60005/13 pin and connector socket for any obvious issue such as "spread socket", broken/ cracked Pin #13, solder failure where pin attaches to PCB, etc. You could test the Module "in situ", without removing it from the vehicle, and see WHERE on the PCB (how close to Pin #13) there is continuity to Chassis Ground when Ignition is turned ON. Remember, DME switches a GROUND signal when Ignition is ON.
b) If a switched Ground signal at or near Pin #13 is identified, then you can either:
(i) Repair,/ have an electronics shop repair, whatever is causing the open circuit between the Board & Pin#13, or
(ii) If Pin #13 is physically BROKEN or irreparable, you can solder a "Jumper" to Pin#12 which is adjacent to Pin #13. Pin #12 is NOT used on any MSV70, MSV80 (your DME), MSD80 that I have seen a Pinout for, and is clearly "NOT used" on your 2012 328xi per the attached Pinout. You do a jumper at the DME Pins, AND a Jumper between Sockets 12 & 13 in the Connector and normal function is restored. Several folks have reported success doing that. I have Personally NEVER OPENED my E-Box.

Hope to learn something from your efforts. Photos would be helpful to the rest of us.
George