Yes, they go right back to high beams (thank God! 
) … I’d be more worried about a cop pulling me over for the dims being out and driving around with just the parking lights on momentarily. Of course, once he realizes it’s a Jaguar, he’ll probably just shake his head, knowingly. 
And good thinking about the swap out - I didn’t realize they are the same type relays … I’ll give that a go … 
Do you have a link to them, VK? I’m almost afraid to see the price, though … 
SNG has it for $105.
If you Google “Jaguar dac6902” you should find others.
One night, just a couple of miles from home, my headlights had a Lucas moment. Fortunately it was only a couple of miles, on well lit urban streets, to get back home. But Mr Plod spotted me and followed me back to base. It is actually OK to drive in the UK with only sidelights in such conditions, so they only wanted to make sure I knew that it was something that was legal but not advised. I agreed, and we had a good chat about the vagaries of Jaguar electrics, and the fondness of Top Gear for the XJS - even though they did lose a stig by launching one off the end of a aircraft carrier. Luckily I was able to keep the conversation going and interpose myself between them and the fully lit, road bollard by the side of the garage that I had “borrowed” whilst a university student.
Wow. Not bad - I have seen used ones going for/asking for over that much! 
It’s amazing what Google can do.
Yep. Looks like the only other new one on there is from another one of the usual suspects (BPP) for $175.00. What’s kinda weird is that particular part is the “OEM” version. It looks like they have another version for sale too (presumably aftermarket) but they want $20.00 more for it. wth? I don’t think I’ve ever encountered a situation where a Jag OEM part costs LESS than its aftermarket version. 
Definitely check the relays under the hood first. You can find out if it’s a bad relay, if there’s power to the low beams, and if the module sends the signal to relay when switching. Great place to start. All located in one spot under the hood.