Cardinality of strongly cellular-Lindelöf first-countable Hausdorff spaces

Determine whether every strongly cellular-Lindelöf first-countable Hausdorff space has cardinality at most the continuum.

Background

The paper shows that strongly cellular-Lindelöf first-countable Urysohn spaces have cardinality at most the continuum. Whether this extends to Hausdorff spaces (without assuming Urysohn) remains unsettled.

This question is closely related to the fundamental open problem on cellular-Lindelöf first-countable spaces and seeks to clarify the impact of separation axioms on cardinality bounds.