Convexity and reciprocal concavity of the coupon-collection objective function

Let $N\in\mathbb{N}$ and, for positive $x_1,\ldots,x_N$, define

$$p_N(x_1,\ldots,x_N)=\int_0^1\left(1-\prod_{i=1}^N(1-t^{x_i})\right)\frac{dt}{t}.$$

Convexity and reciprocal-concavity conjecture. For each $N\in\mathbb{N}$, the function $p_N$ is convex; indeed, $1/p_N$ is concave. This claim concerns the network objective function arising in coupon collection and predicts a generalization of the verified low-dimensional partial-fraction patterns.