Given a convex plane domain, the billiard ball map sends the
$\def\R{\mathbb{R}}$
incoming ray (the trajectory of the billiard ball) to the outgoing
one: the law of reflection is “the angle of incidence equals the
angle of reflection”.

Consider two nested convex domains.
Then one has two billiard ball maps, $T_{1}$ and
$T_{2}$, acting on the oriented lines that intersect
both domains. If the domains are bounded by confocal ellipses, then the respective billiard ball
maps commute; see, e.g.,
Tabachnikov [2005]
.

Assume that the two maps commute: $T_{1}\circ T_{2}=T_{2}\circ T_{1}$.

######
Conjecture 1.

The two domains are bounded by confocal ellipses.

For outer (a.k.a. dual) billiards, an analogous fact is proved in
Tabachnikov [1994]
.
For piece-wise analytic billiards, this conjecture was proved by
Glutsyuk [2014]
.

Of course, this problem has a multi-dimensional version, open both for inner and outer billiards.