Wu's second asymptotic conjecture for binomial sums

Let $m$ be a positive integer, and let $\gamma$ denote the Euler–Mascheroni constant. Wu's second asymptotic conjecture.

$$\sum_{k \geqslant 1} (-1)^{k}\binom{m}{k} \log{k} \sim \log{\log{m}} + \gamma \quad \text{as } m \to \infty.$$

This is one of three asymptotic conjectures used to reduce regularised irreducible loop integrals to computable forms; its resolution is not specified in the source.