Answer: Brown rot fungus can be a problem on nectarines, and symptoms on the fruit are as you describe. Upon inspection of your trees, you'll probably find other signs, as well. The fungal spores survive on diseased twigs and old, rotten fruits and the spores spread by air currents, rain splash and insects. There's a gummy ooze associated with the infection, and affected twigs and leaves shrivel and die early in the season.
To help control this disease, prune and destroy affected twigs and branches. This pruning will also help open the plant up to good air circulation throughout the canopy. Avoid wetting blossoms, foliage and fruit. Be sure to rake and destroy all fallen fruit and leaves at the end of the season.
You can apply a Bordeaux mixture (or other copper-containing fungicide), at budswell to help protect the leaves and blossoms from infection. Apply again after the leaves appear.
With preventative sprays and good garden sanitation you should be able to keep brown rot under control.
I cannot confirm that Asian pears are susceptible to the same brown rot fungus. The foliage and fruit of these trees are generally pest-free. I'd concentrate my efforts on controlling the problem with the nectarine tree with the expectation that the Asian pear fruit will benefit from the absence of the fungus in a nearby tree.
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