An upper bound to second order for the longest minimum-weight path in a complete graph

Give the edges of the complete graph on $n$ vertices independent exponential weights and let $M_n$ be the largest number of edges of a minimum-weight path between two vertices. Addario-Berry, Broutin and Lugosi proved that $M_n/\log n\to\alpha$ in probability, where $\alpha\approx3.5911$ solves $\alpha\log\alpha-\alpha=1$, with the bounds $\alpha\log n-L\log\log n\le M_n\le\alpha\log n+\omega(n)$. We prove that $\mathbb{P}(M_n\ge b_n+x)\le C\alpha^{-x}$ for all $x\ge0$, where $b_n=\alpha\log n-(\log\log n)/(2\log\alpha)$. Hence $M_n-\alpha\log n\to-\infty$ in probability and $\mathbb{E} M_n\le b_n+O(1)$. We also find the asymptotic value of the expected number of pairs of vertices joined by a minimum-weight path of at least $k$ edges when $k=\alpha\log n+O(\log\log n)$, show that at $k=\lceil b_n\rceil$ the second moment of this number tends to infinity although its mean stays bounded, and prove that $\mathbb{P}(M_n=n-1)=n/(2\,(n-1)!)$ for every $n\ge2$. The lower side of the question of van der Hofstad on the fluctuations of $M_n$ stays open.