It's not likely that the programming for your car will ever allow the
cold start valve to operate. If it does however it only fires when the
starter is engaged.

Your problem sounds as if you are losing fuel pressure. You'll have to
hang a gauge in the system to eliminate the possible causes. You need to
know if the fuel pressure regulator, main fuel pump check valve, or
prepump hose or prepump itself are allowing the fuel pressure to drop.
It's also possible that the injectors themselves are not closing
completely on shut down and are dripping or the fuel pump relay is
intermittent and the pump isn't even running until the starter has been
spinning for a while. Instead of just starting the car immediately, try
cycling the key a few times and listen for the fuel pump to run for a
second or two to prime the fuel lines. If this cure the problem then
proceed back to the beginning of the paragraph.

If the above doesn't provide a solution then perhaps some ignition
maintenance is in order. It is possible that the coil is intermittent
and provides a weak spark when the trouble occurs, compounded by well
worn spark plugs or old wires, cap and rotor.

This general information. For a more specific response post the fuel
injection and ignition types. Depending on type there are additional
components that may be faulty.

Bob