Equal-probability conjecture for the expected sibling deficit

There are $N$ coupon types, with draws made independently according to a probability vector $\mathbf{p}=(p_1,\ldots,p_N)$ in the open simplex

$$\mathcal{P}_N=\{\mathbf{p}\in\mathbb{R}^N:p_k>0,\ \sum_{k=1}^N p_k=1\},$$

and let $T_N$ be the first time every type has appeared. For an integer $j\geq 2$, define

$$U_j^N=\#\{k\in\{1,\ldots,N\}:\text{type }k\text{ is drawn fewer than }j\text{ times by }T_N\}.$$

Let $\mathbf{u}=(1/N,\ldots,1/N)$ be the uniform probability vector. Equal-probability conjecture. For every $N\geq 2$ and every integer $j\geq 2$, the map $\mathbf{p}\mapsto\mathbb{E}[U_j^N]$ on $\mathcal{P}_N$ attains its maximum at the uniform vector $\mathbf{u}$, and only there. This asserts that equal coupon probabilities uniquely maximize the expected deficit in the siblings model; the surrounding discussion describes the equiprobable distribution as extremal, while the conjecture remains unresolved in the supplied source.