Polyhedrality for twisted sums with $C(ω^α)$

Abstract: We obtain two partial answers to the 3-space problem for isomorphic polyhedrality: (1) every twisted sum of $C(\alpha)$, $\alpha<\omega_1$, with a separable isomorphically polyhedral space with the BAP, is isomorphically polyhedral. (2) Every twisted sum of $c_0(\kappa)$ and a Banach space having a boundary with property $()$ has a boundary with property $()$, hence it is isomorphically polyhedral.