I dunno where the reverse gear sensor lives but I suspect it's somewhere where it'll get covered in dirt grease and oil, so check that the connector to the switch is clean, connected and "live" (actually has current). It reads like yours does.
If there's current to the switch and the switch is good then the current is just not reaching the lamps...
You could start at the lamps and work backwards until you find the current.
( The lamps are not in the bootlid are they? If so, the hatch wiring loom inevitably snaps a few wires over the years) but if the bulbs are good and the wiring to them is good, the most likely place where the current to both of them can go AWOL at the same time is at the fusebox (under the "right knee" cover) so I would look there first.
Additionally... Stilo is CANBUS and has a Body Control Module behind the fusebox. You'll need someone with a wiring diagram.. but if the reverse lamp signal goes via the BCM then the fault could be inside the BCM (circuit board).
BCM is not just a toss it away and fit a new one job unfortunately... it contains the chips for the immobilisor so you would need to repair it.
If it's not repairable, you could swap it for a "new" one, as long as you also swapped over the chips... or (since you'll need professional help probably and it's going not be a DIY fix) you could instead leave it alone and spend the money on an auto-electrician to wire the reversing lights old style/outside the BCM.
But fuse-box first.
Ralf S.