The extremal Windsor-move conjecture for Candy Nim games

For a Candy Nim game $G$, let $N(G)$ be its total number of candies and let $N_W(G)$ be the number of candies Windsor takes under optimal play. Fix $N>0$.

Extremal Windsor-move conjecture. There exist, not necessarily distinct, games $G_1$ and $G_2$ with $N(G_1)=N(G_2)=N$ such that

$$N_W(G_1)=N_W(G_2)=\max_{H;\,N(H)=N}N_W(H),$$

where $G_1$ has a pile containing at least $N/4$ candies and $G_2$ has at most $c\log N$ piles for some absolute constant $c>0$.

This conjecture describes two structural forms of games attaining the maximum possible number of candies taken by Windsor among games with fixed total size: one with a large pile and one with logarithmically many piles.