Agreed with ||whiteout||. There are two problems here.
1. Tenants shouldn't be walking on carpet with their shoes, especially not in a common area. From another angle, let's say it's not carpet and lino, should other tenants have to deal with the basement tenants tracking snow, ice and water on the stairs to access the shared laundry area?
2. Linoleum to match top and bottom probably would be cheapest and easiest to install. If you're concerned about slipperiness of linoleum, consider installing stair nosing?
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