Polyhedrality for twisted sums with $C(\omega ^\alpha)$

We obtain two partial answers to the $3$-space problem for isomorphic polyhedrality: (1) every twisted sum of $C(\alpha )$, $\alpha \lt \omega _1$, with a separable isomorphically polyhedral space with the BAP, is isomorphically polyhedral. (2) Every twisted sum of $c_0(\aleph )$ and a Banach space having a boundary with property $(*)$ has a boundary with property $(*)$, hence it is isomorphically polyhedral.