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In GSA Pro there is a switch to turn openstreetmap from colour to grey:
is there any possibility for GSA Server / Lite, to do so also, i.e. via serviceprovider.xml or somewhere else?
There is no such switch in Lite unfortunately but there are greyscale openstreet map tile providers out there. The one I know is this one https://maps.omniscale.com/en/ They have a fair use policy (just like OSM). Maybe you can give that a try.