Circularly chainable continua that annularly inscribe squares

Let A denote the annulus $\{x\in R^2:1\le |\left|x\right||\le 1+\sqrt{2}\}$. In this paper we prove that if X is a circularly chainable continuum essentially embedded in A, then X contains the four vertices of an Euclidean square of side length at least $\sqrt{2}$. Furthermore, let us say that a plane continuum X satisfies the property of annular inscription in A if whenever X is essentially embedded in A, the image of such embedding admits an inscribed square of side length at least $\sqrt{2}$. We show that every circularly chainable, not chainable plane continuum satisfies the property of annular inscription in A and, moreover, that the property of annular inscription in A is generic among continua that separate the plane.