While searching the forum today, I realized I never posted an answer to Danw2's excellent response in my thread on flame rods:
thread698-285700: Flame rods in coker furnace

The cabling changes constituted a change from original wiring and did work for a while, but problems recurred after awhile with the furnace in service.

The coaxial shield is being used to provide a ground path for the flame rod, and did not prove to be the issue.

Some flame rod tips proved to be fouled with flame corrosion/soot accumulation; cleaning and replacing corrected these.

The five rods furthest from the fireye relays were cabling faults - the conduit containing the coax had been run too close to a heat source, thus causing the cable to fail.  The conduit was rerouted, insulated, and the cable repulled.  All have been in service for nearly a year with no further issues.

Thanks again to Dan for his input!