Looking at the pic, I don't see any evidence sprinkler mineral build-up. What I do see are pavers that have shifted due to loss of sand in the joints. The joints are filled with "gunk". A thorough cleaning with a surface cleaner along with a turbo for trimming should clean it right up.
What you should propose - and advise them it will need to be done after the cleaning - is re-sanding the joints and an application of a sealer. I would think that this walk would be an excellent candidate for Seal 'n Lock - it's a two-part water-based polyurethane that will lock the sand in the joints and bring back the color of the pavers. The folks at Seal 'n Lock say that $1 to $1.50 per sq ft is appropriate. Price includes cleaning, sanding (using your pwer) and sealer. Since it is water-based, it can all be done in one day with no milky haze from moisture.
P.S. Don't forget to kill the weeds before you seal!