There's two possible issues, we we need to do two different tests to see what problem is. Might be both areas are bad.
TEST FUEL CUT RELAY
Bench-test relay as above. This will give 100% verification relay is good or bad.
Solid-state relays are direct drop-in replacement for mechanical types and operate exactly same way.
Replace coil with pnp-junction of transistor and it flips switch same way.
Relay has no idea if ground signal is coming from ignition-coil or from ground-jumper.
TEST IGNITION SIGNAL AND CIRCUIT
The ignition signal is a ground. Digital electronics has three possible states: high, low or open.
Ignition signal on R/Y wire is either open or low. When it's low, R/Y is connected to ground to dump coil.
Ignition circuit provides the ground for relay's coil while black provides power.
If you don't have proper ground coming from R/Y terminal, then that wiring to coil may be bad.
Can test grounding pulses with solenoid-light. https://www.amazon.com/dp/B004FEN84M/ Connect one terminal to black wire and other to R/Y wire at fuel-cut relay connector. Also test for proper grounding pulses by connecting to ignition coil. It may be case of bad wiring and signal is present at coil, but not at relay-connector.
Or test with oscilloscope:
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The grounding pulses are extremely short. If R/Y wire between relay and coil has drop-out like black wire, it's very possible these ignition-pulses aren't long enough to trigger relay and keep it activated. To verify wiring, test signal at ignition-coil itself, then compare trace to one at relay-connector. If they're not identical, wiring is bad between coil and relay-connector. You can replace that R/Y wire end-to-end. Also test other coil's trigger signal with oscilloscope as well.
"Fuel cut relay" is really bad terminology, might be translation issue like "all your bases are belong to us". Before relay can cut power to pump, it must first provide power to pump like any other normal relay. On many bikes and cars, this relay is wired into crank-sensor so that relay doesn't activate fuel-pump unless engine is spinning. This comes from 1992 FMVSS regulation that dictates some safety requirements. Tying fuel-pump operation to engine-speed is one way of satisfying those safety-requirements. Prior to this, you can be trapped upside-down in flipped-over car after crash and fuel-pump can merrily continue pumping and dump entire contents of tank from ruptured fuel-lines onto hot exhaust-pipes!