EDIT: now generalized and asked at MO.
This is an outgrowth of an earlier MSE question, which itself was motivated by an MO question. An answer to this question is claimed there as well, but the details are insufficient for me at present, so:
This is a question about "columns-only $p$-color Hackenbush." Formally, for a natural number $p\ge 4$, a ($p$-)board is a finite formal sum (or multiset if one prefers) of finite strings from $[p]=\{1,2,...,p\}$. "Colors" (elements of $[p]$) correspond exactly to players. For $i\in[p]$ and a board $B$, a legal move in $B$ for player $i$ consists of replacing one occurrence of a string $\sigma\in B$ with one occurrence of a string $\tau$ such that $\tau i\preccurlyeq\sigma$. As in the above-linked questions, define the "basic type" of a board $B$ to be the set $bt(B)$ of pairs $(x,A)$ such that $x\in [p]$, $A$ is a nonempty proper subset of $[p]$, and there are strategies $\pi_a$ for $a\in A$ which when followed ensure that the first player to be unable to move will not be in $A$ assuming player $x$ starts and move ordered proceeds in the obvious cyclic order from then on.
Intuitively, a string $\sigma$ occurring in $B$ represents a "stack of edges" labelled according to $\sigma$ from the ground up, and $(x,A)\in bt(B)$ means "team $A$ can avoid losing in $B$ if $x$ goes first."
Now let ${\bf z}_p=\langle 1\rangle+\langle 2\rangle+...+\langle p\rangle$ be the "rainbow" board with $p$ colors. Interestingly, and in contrast with the two-color version, adding ${\bf z}_p$ does not generally preserve basic type (see below). Despite this, I'm curious whether adding ${\bf z}$ does eventually preserve basic type:
Fix $p\ge 4$. For a board $B$, is there necessarily a natural number $k$ such that $$bt(B+m{\bf z}_p)=bt(B+n{\bf z}_p)$$ whenever $m,n\ge k$?
Here $aC=C+C+C+...+C$ ($a$ times) for $a\in\mathbb{N}$ and $C$ a board.
To motivate the constraint on $p$, consider the $p=2$ and $p=3$ cases of the same problem. When there are only two players, we're just looking at Red-Blue Hackenbush and "adding ${\bf z}_p$" to a game never affects the basic type of a game. When $p=3$ this is no longer the case - consider e.g. $\langle 123\rangle$ vs. $\langle 123\rangle +{\bf z}_3$, with players $1$ and $2$ on the same team and $1$ moving first - but the situation is still pretty simple: adding ${\bf z}_p$ to a game can only make things better for a coalition consisting of all but one player, and that's all the coalitions there are when $p\le 3$. For $p\ge 4$, though, things seem more complicated (e.g. consider $A=\{1,3\}$). On the other hand, I don't yet have an example of a $p$-board $B$ where the "basic type sequence" $$\mathfrak{Z}(B):=\langle bt(B+k{\bf z}_p)\rangle_{k\in\mathbb{N}}$$ ever changes value more than once, so things might wind up being the same as with $p=3$.