Equality of the tail constants for maximal displacement

Let $M$ be the whole maximal displacement of the critical branching random walk in random environment, and suppose that finite positive constants $C_1$ and $C_2$ satisfy

$$C_1\leqslant\liminf\limits_{x\rightarrow\infty}x^{\frac{2}{3}}\mathbb{P}(M>x)\leqslant\limsup\limits_{x\rightarrow\infty}x^{\frac{2}{3}}\mathbb{P}(M>x)\leqslant C_2.$$

Equality of the tail constants. The constants $C_1$ and $C_2$ are equal. This would sharpen the established two-sided asymptotic-order estimate for the tail probability of the whole maximal displacement. The supplied text does not indicate whether this claim has been resolved.