  We prove that every $C^*$-embedded subset of $\ss$ is a hereditarily Baire
subspace of $\mathbb R^2$.
  We also show that for a subspace $E\subseteq\{(x,-x):x\in\mathbb R\}$ of the
Sorgenfrey plane $\mathbb S^2$ the following conditions are equivalent: (i) $E$
is $C$-embedded in $\mathbb S^2$; (ii) $E$ is $C^*$-embedded in $\mathbb S^2$;
(iii) $E$ is a countable $G_\delta$-subspace of $\mathbb R^2$ and (iv) $E$ is a
countable functionally closed subspace of $\mathbb S^2$.
