Ballistic deposition height-growth conjecture on a strip

Let $(X_n)_{n\in\mathbb N_0}$ be independent and identically distributed random variables sampled uniformly from $[K]$, and let $H_n:[K]\to\mathbb N_0$ be the height profile of the ballistic deposition process, initialized by $H_0(k)=0$ and updated according to the strip deposition rule. For each fixed $K$ and $j\in[K]$, write $H_n(j)$ for the height at location $j$ at time $n$. The notation $a_K\sim b_K$ means

$$\lim_{K\to\infty}\frac{a_K}{b_K}=1.$$

Ballistic deposition height-growth conjecture. With probability one, for all $j\in\mathbb N$,

$$\lim_{n\to\infty}\frac{H_n(j)}{n}=\lim_{n\to\infty}\left(\max_{k\in[K]}\frac{H_n(k)}{n}\right)\sim\frac{4}{K}.$$

The claim predicts a common asymptotic linear growth rate for every site and for the maximum height, with rate asymptotic to $4/K$ as the strip width tends to infinity. The surrounding discussion reports numerical simulations and mentions previously obtained lower and upper bounds, but the supplied text gives no resolution of this conjecture.