Conjectured asymptotic dependence of truncated martingale tails on the starting position

Let $W_\infty^{(x)}$ and $D_\infty^{(x)}$ denote the truncated martingales started from position $x$, and let $\kappa$ be the parameter appearing in the tail behavior of $W_\infty^{(x)}$. For $x,y\to\infty$, the conjectured dependence on $x$ is modeled by the probability that a particle starting from $x$ gets close to the origin. Tail-asymptotic conjecture. Under mild assumptions, for some $K\in(0,\infty)$,

$$\mathbb{P}(W_\infty^{(x)}>y)\sim Ke^{-\kappa x}\mathbb{P}(W_\infty^{(0)}>y),\qquad x,y\to\infty,$$

and similarly, for some $K\in(0,\infty)$,

$$\mathbb{P}(D_\infty^{(x)}>y)\sim Ke^{-x}\mathbb{P}(D_\infty^{(0)}>y).$$

This would give precise tail asymptotics for the truncated branching-random-walk martingales and quantify their dependence on the initial position. The paper presents these asymptotics as an open problem under mild assumptions.