Assuming countable choice, the deleted Tychonoff plank worked as $T_3$ but not normal inside its compact Hausdorff normal parent

Assume countable choice. In $P=(\omega_1+1)\times(\omega+1)$ remove the corner $(\omega_1,\omega)$. The resulting plank is regular and $T_1$, hence $T_3$, but it contains the closed edge sets $\{\omega_1\}\times\omega$ and $\omega_1\times\{\omega\}$ that have no disjoint open neighbourhoods.

The parent is compact Hausdorff and normal. The obstruction is concrete: a supposed separation gives an ordinal bound for the countably indexed vertical-tail neighbourhoods, and a horizontal-tail neighbourhood beyond that bound meets a vertical-tail neighbourhood. All stated properties, including the location of countable choice, are proved in Assuming countable choice, the deleted Tychonoff plank is a regular nonnormal open subspace of a compact Hausdorff normal space.