Pseudo-solenoids are not continuously homogeneous

The topological space X is a pseudo-solenoid if X is a hereditarily indecomposable, non-chainable, and circle-like continuum. It is shown that no such continuum is continuously homogeneous. Specifically, there are two points in X, identified as z and 1¯, such that for any continuous surjection f:X→X, f(1¯)≠z. This answers a question of J.J. Charatonik in the negative.