Faulhuber's rotational invariance conjecture for Weyl–Heisenberg frames

Let $obreak\boldsymbol{\textstyle\frac{}{}} \phi\in L^2(\mathbb{R})$. Write $W\phi$ for its Wigner transform, let $\Lambda$ be a lattice in phase space, let $\mathcal{G}(\phi,R\Lambda)$ denote the associated Weyl–Heisenberg frame system, and let $R\in SO(2,\mathbb{R})$ act on the lattice. A function is a Hermite function if it is one of the Hermite functions in $L^2(\mathbb{R})$.

Rotational invariance conjecture. The following are equivalent:

1. $\phi$ is a Hermite function.
2. $W\phi$ is rotation-invariant.
3. The frames $\mathcal{G}(\phi,R\Lambda)$ possess the same frame bounds for all $R\in SO(2,\mathbb{R})$.

This conjecture characterizes Hermite functions through rotational invariance of their Wigner transforms and through invariance of Weyl–Heisenberg frame bounds. The one-dimensional rotational invariance of Hermite functions motivates it, but the source gives no resolution of the equivalence.