Some past experience on other vehicles with same problem and gremlins. Try a hard re-set on the system. Remove battery terminals, then short out battery terminals, NOT on the battery itself. Use a piece of suitable wire, not flimsy thin stuff, 14G or above. You need to connect the + and _ minus terminals together, but DONOT touch anything else metal, do not even let the wire you are using if you cannot touch the battery terminals touch anything. This will sometimes allow the system to fully discharge any electric still in the wiring. Do the touching of the battery terminals for about 5 mins.
Then leave battery disconnected for say an hour or more.
Reconnect battery and see if you can then hear sound.
Then if still the same you now know it's something else.
look at, if you are able to the fibre cables that connect the system behind the head unit. You have to get an auto-electrician to do this if you are not able. Normally the fibre optic cable gets damaged which if you don't know how to check this, then electrician will be needed. If cable is fine on both channels then you have to search further. A bad wire or rusty wire is the next line to go down.
You need to trace the wiring through the car from the speakers right back to your head-unit. They should show perfect current with-out even a hint of electrical flow. with-out having stripped down this model of car, you will have to find and locate the wires yourself.
When all wires are definitely 100% guaranteed to be working, not a single chaff or split or pinch anywhere, and every earth wire is making an absolute 100% rustless connection to the body then you know this part is done.
If still the same, next move onto the head-unit itself and if memory my end is correct you have amp units somewhere that amplify the sound, this is then also got to be checked, i.e. removed and tested by auto-electrician.
All this can be time consuming as you will have to remove panels etc, to get access to the wiring which runs from speaker to speaker and back to amps, them back to head unit.
Also check out your actual ignition barrel as this can cause a lot of gremlins in many cars as the electrical side of the ignition barrel gets grubby over years and springs etc, inside do funny things like cause shorting out.
Also try this but not sure if a Jag would suffer this but free to try. Turn off the system BEFORE you turn off the ignition.
Turn on the ignition and engine BEFORE you turn back on your system.
A hard-flash update from Jaguar to your system can also be a possibility as this can sort out a lot of problems across the board in all car manufacturers. Car auto-electricans are not that expensive but if they have to strip panels out and chase wires then the hours can mount up.
The actual fibre harness, if indeed this is what Jag have fitted to these models can also have to be replaced and this can be costly.
For what you describe I would not worry about spending a few hundred £'s, say 2-300 to have your system fully working again through process of elimination. If units themselves are found to be the cause, look for second-hand ones.
Car audio auto-electricans are where to look, but NOT back street garages, you need fully intelligent and qualified people to do these things and not someone who is blundering through in the hope they MAY discover something. Seen to many supposedly trained mechanics who are dangerous to themselves and certainly to others.