Let $C_n(\rho)$ be the $n$th cumulan of the probability density $\rho$.
If $C_n=0$ for $n>1$, then $\rho$ is a $\delta$ function.
If $C_n=0$ for $n>2$, then $\rho$ is a gaussian function.
Do we know the general form of probability densities with $C_n=0$ for $n>3$?
The statement that $\rho$ must be gaussian if $C_n=0$ for $n>2$ is not true in general. This is only true if the support is the real line.
As per your question, you can maximize the entropy $S[\rho]=-\int \rho\log\rho dx$ with the constraints that the first cumulants are known. The solution to this problem will again depend on the support.