Is it permissible to place clean, wet clothes on clotheslines that had ritualistically unclean clothes on them, and then dried by the sun and air, without washing these clotheslines?

Ropes do not usually absorb impurities, and clothes placed on them remain pure after washing and wringing. If they do absorb a small amount of impurity, it dries quickly, and its traces disappear with the sun and wind. Therefore, there is no need to be overly scrupulous in avoiding them.