You should definetly convince your client to upgrade. the 2.7x branch of Limesurey was one of the worst versions of Limesurvey, in my opinion.
The installation 2.73.0-build171219 is from the end of 2017. This means this platform hasn't received updates and bug fixes for 2.5 years, which is quite a long time for web applications.
He is using it for several years and is comfortable with this one
I totally understand. But a lot of people were comfortable with Windows XP, but they had to move on...
But if you update, don't go to LS 4.x just yet. It is not ready for production in my opinion. Currently the best and most stable LS version is LS 3.x.