I'd check FICM voltage first. Check cold while KOEO, cranking, running, and high idle. If that checks I'd get it to run and drive, then re-check at operating temperature. If that checks out it 90% rules out the FICM. Then I'd say your issue is most likely an HPO leak, which gets worse when the engine's warm as things expand. The IPR read can help diagnose if that's the issue, but you need the right tools to do it, so it may require a trip to a shop. You can also air test the HPO system with the right tools. ICP is another possibility; you can unplug it and see if the truck will start, that's sort of a backwoods way to test it, but effective nonetheless. The issue is that on an '04 it's under the turbo, so hard to get to, which is why reading its duty cycle is a good way to test without having to take anything apart.
'06 F250 4x4 - 5" Flo Pro exhaust, SB Filter intake, Accufab elbow, Edge Evolution (monitoring only), SCT w/ ID custom tune, FASS 195 pump, Gillette Diesel EGR cooler delete, Sinister Diesel coolant filter, ELC coolant, updated turbo drain tube/oil feed line/STC fitting/oil cooler, ARPs w/OEM HGs, Elite coolant lines, ITP RR fuel system, RCD 175/30 injectors, Powermax, BD CCV, FICM.com FICM w/ ID tune, Elite UP, BPD water pump - 13.069 @ 101.94
'02 WRX - Outback rear disc swap, EBC green pads, DBA pillar vane rotors, TXS UP/TBE/TMIC, Perrin LW crank pulley, PPG billet steel shift forks, ACT Streetlite flywheel & clutch, K&N filter, STi Group N motor/trans mounts, TiC/Kartboy rear diff mounts/subframe lock bolts/outrigger stiffeners, Kartboy SS & all shifter bushings, custom PDX tune for Cobb AP - went 14.1 on a terrible 60ft before most of these mods; shooting for 13.50s