The square-exponential tail conjecture for the excited random walk's maximum

Let $V(t)$ denote the random variable measuring the quantity studied in the preceding estimates for the excited random walk at time $t$, and let $C,c>0$ be constants. Square-exponential tail conjecture. The correct tail decay is square-exponential, namely

$$\mathbb{P}(V(t)>\lambda\sqrt{\log t})\leq Ce^{-c\lambda^{2}}.$$

The surrounding discussion notes that the paper only obtains weaker stretched-exponential estimates and that a sharper estimate of this form would improve the tail analysis of $V(t)$; the conjecture remains unresolved in the supplied text.