Full range of game values in infinite chess

For infinite chess, let $\omega_1^{\mathfrak{Ch}}$ be the supremum of game values for positions with at most finitely many pieces, let $\omega_1^{{\mathfrak{Ch}},c}$ and $\omega_1^{{\mathfrak{Ch}},\mathrm{hyp}}$ denote the corresponding canonical and hyperarithmetic bounds, let $\omega_1^{ck}$ be the Church–Kleene ordinal, and let $\omega_1^{\widetilde{\mathfrak{Ch}}}$ be the supremum for arbitrary positions. Full-range conjecture. The omega one of chess, for both finite and arbitrary positions, is as large as possible:

$$\omega_1^{\mathfrak{Ch}}=\omega_1^{{\mathfrak{Ch}},c}=\omega_1^{{\mathfrak{Ch}},\mathrm{hyp}}=\omega_1^{ck}\qquad\qquad\omega_1^{\widetilde{\mathfrak{Ch}}}=\omega_1.$$

This asserts that chess realizes the maximal game-value ranges allowed by the relevant computability and set-theoretic bounds; the supplied text gives no resolution.