Conjecture on exceptional Gaussian measures for Euler-flow non-invariance

Let $\mathcal{M}^{\sigma}$ denote the class of Gaussian measures considered in the paper, parametrized by sequences $(a_n)_{n\in\mathbb{Z}^2}$, and let quasi-invariance mean quasi-invariance in the sense of the paper's stated assumption under the two-dimensional Euler-vorticity flow. The exceptional sequences are those supported in a line of $\mathbb{Z}^2$ through the origin or on a circle centered at the origin.

Exceptional-support conjecture. Part (i) of the theorem on invariant measures already contains essentially all counterexamples to non-invariance: for $bc=\mu_{(a_n)}\in\mathcal{M}^{\sigma}$, with $c3>0$ (or possibly $c3>-1$ if a suitable local well-posedness theory can be established), $bc$ is quasi-invariant under the Euler flow if and only if $(a_n)_{n\in\mathbb{Z}^2}$ is supported in a line through the origin or on a circle centered at the origin.

This conjecture identifies the geometrically exceptional Fourier supports as the only source of quasi-invariant Gaussian measures in this setting. The paper's results establish non-invariance outside the stated exceptional configurations, while the asserted classification beyond the proved examples remains conjectural.