(455/1023,456/1023) | (221/1023,222/1023)

The two parameters are centers of primitive hyperbolic components (the Julia set looks like Kokopelli).

This example, communicated by Adam Epstein, has a long ray connection, of diameter 8: [221, 222, 226, 217, 806, 71, 966, 934]

You can download two versions, one is a zoom on part of the sphere.