That crud might be a very important clue. Have a look at the fuse that supplies power to the Quartet Harness. This is Fuse 'F', in the main fusebox (it supplies power that is switched on when the ignition is turned on, referred to as 'terminal 15' power). Fuse 'J' also supplies power to the Quartet Harness, but it is continuous power (terminal 30 power), and because the heated grips do not function with the ignition off, it is unlikely that Fuse 'J' supplies them with power.
Unfortunately, the wiring diagram in the Honda service manual does not show the pinouts of the Quartet Harness - it only shows the connector that the Quartet Harness plugs into, which is labelled "Option". Fuse 'F' supplies terminal 15 power via the left-most wire track (on the wiring diagram, not the actual motorcycle!) leading to the "Option" connector on the wiring diagram. Fuse "J" supplies terminal 30 power via the second wire from the left.
Fuses F and J are the two fuses that run 'sideways' in the fuse block. In other words, there are 5 fuses in a row, and two others that are oriented 90° differently. You want to check those two.
Michael