Doumas–Papanicolaou's finite variance-minimization conjecture for the double Dixie cup problem

Let $m$ and $N$ be fixed, let $p$ range over the probability simplex on $N$ coupon types, and let $T_m(N)$ denote the number of draws needed to collect at least $m$ copies of every type. For a probability vector $p$, write $\operatorname{Var}_p(T_m(N))$ for the variance under the corresponding coupon-collection process, and let

$$u=(1/N,\ldots,1/N).$$

Finite variance extremality. The equal-probability vector $\nu$ minimizes

$$p\mapsto \operatorname{Var}_p(T_m(N))$$

over the probability simplex. The conjecture concerns variance minimization in the nonuniform double Dixie cup problem. Its $m=1$ case was later proved by Yu, while the statement for general fixed $m$ remains unresolved in the supplied source.