L1 convergence for greedy lattice paths

Let $\{X_{v}:v\in\mathbb{Z}^d\}$ be i.i.d. random variables, and let $M_n$ denote the maximum weight of a lattice path of length $n$ from the origin. Assume that there exists $\alpha>0$ such that

$$E(X_0^{+})^d(\log^{+}X_0^{+})^{d+\alpha})<+\infty$$

and that

$$\int_{0}^{+\infty}P(X_0< -t)^{2d}\,\mathrm{d}t<+\infty.$$

$L^1$ convergence conjecture. There exists a constant $M\in(-\infty,+\infty)$ such that

$$E|M_n/n-M|\overset{n\to\infty}{\to}0.$$

The stated negative-tail integrability condition ensures that $EM_n>-\infty$; the conjectured $L^1$ convergence for greedy lattice paths is presented as an open problem requiring new ideas beyond the lattice-animal argument.