I'm not a specialist on the XJ40 but it looks like a well-known problem.
Have you removed heat-shields from the engine or have you put insulation
under the bonnet leading to overheating the fuel-lines? Is it correct that
after cooling down everything is OK?
In that case, try driving for a testperiod, which normally causes stalling,
with the bonnet opened up. That worked for my old cars that had a problem
with it.
Fred

check fuel fault 44...oxygen sensor....peter j

The fuel pressure regulator at the end of the fuel rail is subjected to heat
expansion and fuel pressures get inadequate at idle.  I learned a hard
lesson to get this knowledge.  My 1988 XJ6 VDP had the same dilemma when
coming to a stop and the engine dropped to idle speed.  I followed the
advice of the Haynes automotive manuals suggestion to install a second fuel
pump.  I piped it into the blank fitting on the front of the fuel rail.  Two
weeks later an overpressurized fuel line burst while I was driving on
Interstate 95 and sprayed raw gas into the engine compartment and ignited an
engine fire which destroyed the car in minutes after the line burst.  Just
as you have experienced, even the experts didn't see this one coming.  I've
handbuilt custom intakes and fuel systems for carbureted, injected,
turbo-charged and superchaged engines for classic 6 cylinder Chevy's and a
couple of 70's era Kawasaki 4 cylinder superbike motors and I've never came
across a fuel delivery system as complicated as the on the 1988 Jag that
burned or my current cat, an '89 XJ6.  Do you hear what sounds like a
tapping rhythm a bad valve lifter would make, somewhere near the firewall
when you come to a stop and are at idle?  You can only hear it when your
inside the car.  The sound is similar to "water hammer" in a piping system
when you shut off a valve in a system without a damping standpipe.  Same
thing is happening to the fuel pressure regulator. When the injectors click
shut,  a good regulator will absorb the back pressure created in the fuel
delivery cycle without losing synchronization to the ignition timing.  When
the diaphragm in the regulator expands from heat or ages, the regulator
loses synchronization at high fuel pressures which occur at idle.  Once the
engine load is off idle, the fuel back pressure drops dramatically to supply
the now hungry injector flow and the regulator functions adequately.  That
explains the "experts" suggestion to find an ignition timing fault.  The CPU
controls injector timing based on feedback from the ignition sensors and
fuel pressure sensors. The back pressure leak through the regulator changes
the fuel delivery volume and the now fuel delivery volume tells the CPU a
false value and it corrects with an ignition timing adjustment and the fuel
pressure at idle is inadequate to open the injectors.  Sort of like a leaky
heart valve in a human. This is what I believe caused the burst fuel line in
my '88.  I was at highway speed, (for my '88  that was usually between 85
and 105mph) I let off the accelerator and an instantaneous but brief
backpressure in the fuel lines from the air flow snapping  shut, told the
CPU to reduce injector flow.  At that instant both fuel pumps were
delivering high volumes through the regulator and the backpressure leak blew
the dual pumped fuel rail line.  Have you ever seen a "top fuel" nitro car
blow a supercharger off the engine on a burnout? Same physics here.  Bad
fuel delivery timing and back pressure into the fuel lines. Nitro burners
react violently to lean fuel to air ratios !!!