The conjecture on semiclassical measures of long-living open-baker eigenstates

Let $B_{\vr}$ be an open baker's map, and let its quantization be the one described in the source. Consider long-living eigenstates and their semiclassical measures. A long-living eigenstate is called non-diffractive when its weight on $D(B^{-1})$ and $B^{-1}(D(B^{-1}))$ is negligible in the semiclassical limit.

Open-baker semiclassical-measure conjecture. All long-living semiclassical measures are supported on $\overline{\Gamma_+}$, and almost all long-living eigenstates are non-diffractive. The corresponding semiclassical measures are then eigenmeasures of $B_{\vr}$.

The claim is motivated by numerical Husimi observations: concentration is seen on parts of the relevant trapped-set geometry, while diffractive components may occur for exceptional states. The assertions are explicitly presented as not proved by the theorem discussed in the source.