Conjecture for the locus of X1 (incenter) over Poncelet triangles: a conic iff the pair is confocal

While researchers (Olga Romaskevich, Ronaldo Garcia) have proven that for Poncelet triangles interscribed between two confocal caustics (elliptic billiard) the locus of the incenter X1 is a conic (see [1] for bibliography), we conjecture that this can only be the case if the pair is confocal. Proofs?
[1] D. Reznik, R. Garcia, J. Koiller, "Can the Elliptic Billiard Still Surprise Us?",
Mathematical Gems and Curiosities ,Math. Intellig. v. 42, Sergei Tabachnikov (ed.), 2020, p. 6--17.