Extension problems of real-valued continuous functions

On $C$-embedded subspaces of the Sorgenfrey plane

“…The following problem is still open: Problem 1.3. [12] Does there exist a first countable completely regular space without property (C * = C)? H. Ohta in [11] proved that the Niemytzki plane has the property (C * = C) and asked does the Sorgenfrey plane S 2 (i.e., the square of the Sorgenfrey line S) have the property (C * = C)?…”

On C-embedded subspaces of the Sorgenfrey plane

“…The following problem is still open: 12]). Does there exist a first countable completely regular space without property (C * = C)?…”