Suslin subspace inside a Suslin Banach space

Determine whether every closed subspace \(Y\) of density \(\omega_1\) of a Suslin space \(X_G\), where \(G\) is the comparability graph of a Suslin tree, contains an isomorphic copy of a Suslin space.

Background

A Suslin space is defined as a combinatorial Banach space generated by the comparability graph of a Suslin tree. The analogous tree-theoretic fact is that every subtree of height ω1\omega_1 of a Suslin tree is again Suslin. The authors ask whether an analogous hereditary phenomenon holds for closed Banach subspaces of density ω1\omega_1.

References

Let $X$ be a Suslin space, and let $Y$ be a closed subset of $X$ of density $\omega_1$. Does $Y$ contains an isomorphic copy of a Suslin space?

— Non-separable combinatorial Banach spaces  (2610.01919 - Borodulin-Nadzieja et al., 1 Oct 2026) in Section 8, Trees and geometry of balls, Problem following the Banach-space characterization problem

However, even the case where each $x_\alpha$ is supported by two points seems to be unclear and interesting: let $l_\alpha, r_\alpha \in T_\alpha$ for each $\alpha$ and let $x_\alpha = e_{l_\alpha} + e_{r_\alpha}$. The question is if the space $Y$ generated by $(x_\alpha)_{\alpha<\omega_1}$ contains an isomorphic copy of a Suslin space.

— Non-separable combinatorial Banach spaces  (2610.01919 - Borodulin-Nadzieja et al., 1 Oct 2026) in Section 8, Trees and geometry of balls, discussion following the Suslin subspace problem