How likely it is that low alternator voltage cause Xenon HID headlamps ballast to fail?

It seems that the alternator is producing a bit low voltage, about 13.4V max, while this should be over 14V AFAIK, undercharing the battery.

The left Xenon HID headlamp ballast failed some time ago, so the left headlamp was not working, despite the D1S bulb being fine (switched the bulbs left-right to check). At a workshop they told me the ballast needs replacing. But yesterday also the right headlamp stopped working.

It seems a bit strange to have both ballasts fail without reason.

How likely the ballast failure is caused by the low alternator voltage?

What else could make the ballasts fail like this? I don’t want to invest in new expensive ballasts to have them fail again.

This is a Volvo XC90 year 2003, UK model. I’m not sure if the light are ones which were originally in the car, I’ve been told that originally D2S/D2R bulbs are used in this model, and not D1S. In the future I would like to switch to mercury-free bulbs like D3S.