I have tested with Living DNA but am still trying to get onto Ancestry as my samples keep failing processing. I uploaded my Living DNA raw data to GEDmatch and had a match with another kit at 37.5 cM which I investigated and turned out to be a 4C1R. Her original upload to GEDmatch was from MyHeritage.
But, she has also tested with LivingDNA, and we match at 54.2cM on their site. I don't know, but I assume that Living DNA may be counting smaller segments in their total match length that other sites disregard due to the higher possibility of false positives.
Have you investigated your Living DNA match for any hints of common ancestry? If you can persuade them to upload to GEDmatch and you do the same, you can see a breakdown of the matching segments by chromosome and segment lengths using the comparison tools. If that shows a single fairly large shared segment amongst the total, it's probably worth pursuing, but it could also show that your match is comprised of several much smaller segments. GEDmatch ignore segments less than 7cM by default.
I'm no expert, but from what I've read any single segment of more than 15cM should be considered at least 99% probability of being inherited by descent. So it really depends how your match is made up by segment sizes.
