High-entropy winning-shift conjecture

Fix a finite alphabet $S$. For a subshift $X\subset S^\mathbb{X}$, let $h(X)$ denote its topological entropy and let $W(X)$ denote its winning shift. High-entropy winning-shift conjecture. For every $\epsilon>0$ there exists $\delta>0$ such that, if

$$h(X)>\log_2|S|-\delta,$$

then

$$h(W(X))>1-\epsilon.$$

This predicts that subshifts with entropy arbitrarily close to the maximum possible entropy have winning shifts with entropy arbitrarily close to $1$.