2 problems

Finite-component conjecture. For every $\alpha\in[0,1/2]$, the estimator $\widetilde{\pi}_{n,\alpha}$ has no more than $N=X_{(n)}+1$ components.

Beta-prior conjecture. The appropriate prior for mixtures of Geometric and Negative Binomial distributions is most likely a $\text{Beta}(a,b)$ prior, with $a,b otin (0,infty)$.