growing plants in a conservatory

I have grown a trachcarpus fortunei in a large pot in the conservatory. All was well for 12 months but in the last 2 months the tips of the fronds have died, although the heart of the tree still produces nice looking growth.

I think I had a red spider mite problem, I sprayed the plant with Bug Clear and I cannot see any critters on the plant now. However, the tips of the fronds are still going brown.

Are the mites still there or has the plant a virus? Is there anything that can be done? I have a trachycarpus planted outside in the borders and that looks very healthy.,