In reply to mattc:
I work for an industrial perfume manufacturer. There are a few perfume ingredients, which in their neat form can degrade certain polymers but the dosages will be very small in the perfume, typically 1% or less and the perfume itself is only at around 0.5% in the formulation so by the time a few tiny aerosol droplets of that have got onto the rope, the dilution rate is massive. There will have been so little of any harmful molecules there that I reckon they will have evaporated off the surface before they could do any harm. Just washing it in detergent will expose it to more contact with perfume ingredients and detergent perfumes are designed to be substantive on polymers and natural fibres, so I wouldn't worry.
If you're really concerned ask the manufacturer of the airfreshener for a banded formulation for the perfume and the product then send it to RAPRA, the Rubber And Plastics Research Association. The rest of the airfreshener formulation will be water, surfactant (solubiliser for the perfume) perhaps a little alcohol and some preservative.