Lacasse's conjecture on the PAC-Bayesian combinatorial sum

Let $\xi,\xi_2:\mathbb{N}\to\mathbb{R}$ be defined by

$$\xi(m):= \sum_{k=0}^m \binom{m}{k} \left( \frac{k}{m} \right)^k \left( 1-\frac{k}{m} \right)^{m-k}$$

and

$$\xi_2(m):= \sum_{j=0}^m \sum_{k=0}^{m-j} \binom{m}{j} \binom{m-j}{k} \left( \frac{j}{m} \right)^j \left( \frac{k}{m} \right)^k \left( 1-\frac{j}{m}-\frac{k}{m} \right)^{m-j-k}.$$

Lacasse's conjecture. For every $m\in\mathbb{N}$,

$$\xi_2(m)=\xi(m)+m.$$

This conjecture arose from numerical evidence in the study of combinatorial sums appearing in refined PAC-Bayesian bounds for majority-vote classifiers. Its status is not resolved in the supplied source.