Solutions conjecture for the lognormal star equation

Let $alpha e0$ and let $gamma^2\trianglelefteq2d$ describe the parameters in the lognormal \star-scale invariant setting; $M$ denotes a solution random measure. Solutions conjecture. The possible laws of $M$ are as follows: if $alpha=1$ and $gamma^2<2d$, $M$ is standard Gaussian multiplicative chaos; if $alpha=1$ and $gamma^2=2d$, $M$ has the law of the derivative martingale $M'$; if $alpha<1$ and $gamma^2<2d$, $M$ is atomic Gaussian multiplicative chaos in the dual phase; and if $alpha<1$ and $gamma^2=2d$, $M$ is atomic Gaussian multiplicative chaos in the frozen phase. The source identifies the last two situations as conjectural, while the first two are stated as proved results, so the complete four-case classification remains open.