Confocality criterion for elliptic incenter loci

Let a pair of ellipses admit Poncelet 3-periodics, and let $X_1$ denote the incenter of a member of the family. The pair is confocal when its two ellipses share the same foci. Incenter-locus confocality conjecture. Over Poncelet 3-periodics in some ellipse pair, the locus of $X_1$ is a non-degenerate ellipse, meaning not a point, if and only if the pair is confocal. Experimental evidence suggests that confocality is precisely the condition for the incenter locus to be a non-degenerate ellipse; the general assertion remains open.