The injectors may be clogged. You say the bike was parked for years so even if the injectors are not completely clogged it's a good idea to get them cleaned and flow matched.
2004 S4R most likely still have the 59M ECU. It's established that those are prone to failure. If you can source 5AM ECU from another 2005, 2006 S4R you can swap and see if it's the problem.
If not I suggest you download IAWDiag software and get the cables so you can use the diagnostic port to pull error codes from the ECU. The 5AM ECU was used on all models and therefore you can get used ECU from any Ducati I believe up to Panigale. Panigale does have different ECU. The ECU can be reflashed with S4R map with the IAWDiag software.
Search for GuzziDiag or IAWDiag on
Ducati.ms forums. There are lots of threads. I will look and post the links to the cables you need because there are some cables that work and some that don't.
For used ECU you can do search on e-bay. The 5AM is not that expensive and there are some for sale from people who upgraded to Termis that came with race ECU.
S4R (996) 59M and 5AM Interchangeability
GuzziDiag / IAWDiag Download
USB cable for Triumph Motorcycles TuneECU FTDI FT232RL Chip OBD II KTM Tune ECU
3 Pin to 16 Pin OBD2 Adapter Connector Diagnostic Cable for Fiat Alfa Lancia US
1198 ECU IAW5AM - needs to be reflashed with S4R map. Can be done with the IAWDiag software
S4R 5AM maps on TuneECU.com