The winning-strategy conjecture for Ideal Chomp on polynomial algebras

Let $\bar{K}[x_1,\dots,x_n]$ be the polynomial algebra in $n$ variables over an algebraically closed field $\bar{K}$, with $n\geq 1$. In the Ideal Chomp Game, player A and player B alternately choose proper ideals according to the game's rules, and a player wins when the opponent has no legal move.

Winning-strategy conjecture. Player A has a winning strategy on $\bar{K}[x_1,\dots,x_n]$ for any $n\geq 1$.

This conjecture concerns the open question of determining the winner of the Ideal Chomp Game on polynomial algebras in arbitrary dimension. The preceding examples show winning strategies for player A in several related algebras, but the general case remains open.