Cellular-Lindelof: a common generalization of the Lindelof property and the CCC

All spaces are assumed to be Hausdorff. Recall that a cellular family in the space $$X$$ is a family of pairwise disjoint non-empty open subspaces of $$X$$. The cellularity of $$X$$ ($$c(X)$$) is defined as the supremum of the cardinalities of the cellular families in $$X$$. CCC means that the cellularity is countable.

We say that a topological space $$X$$ is cellular-Lindelof if for every cellular family $$\mathcal{C}$$ there is a Lindelof subspace $$Y$$ of $$X$$ such that $$U \cap Y \neq \emptyset$$, for every $$U \in \mathcal{C}$$.

Clearly every Lindelof space is cellular-Lindelof and every CCC space is cellular-Lindelof.

The cellular-Lindelof property was introduced in our paper with Bella https://link.springer.com/article/10.1007/s00605-017-1112-4, where we note that:

FACT: Cellular-Lindelof first-countable spaces have cardinality at most $$2^{\mathfrak{c}}$$.

Indeed, let $$X$$ be a first-countable cellular-Lindelof space. Then $$c(X) \leq \mathfrak{c}$$ (this follows from Arhangel'skii's Theorem stating that every Lindelof first-countable space has cardinality at most continuum). Combining that with the Hajnal-Juhasz inequality $$|X| \leq 2^{\chi(X) \cdot c(X)}$$ (where $$\chi(X)$$ denotes the character of $$X$$) we obtain that $$|X| \leq 2^{\mathfrak{c}}$$.

QUESTION: Let $$X$$ be a first-countable cellular-Lindelof space. Is $$|X| \leq \mathfrak{c}$$?

A positive answer would lead to a common generalization of Arhangel'skii's Theorem and the Hajnal-Juhasz theorem stating that first-countable CCC spaces have cardinality at most continuum.

EDIT: (04/03/2019)

A partial answer can be found in a new paper with Bella: the answer is yes for normal spaces under $$2^{<\mathfrak{c}}=\mathfrak{c}$$.

Another partial answer can be found in yet unpublished work of Vladimir Tkachuk and Richard Wilson: the answer is yes for "cellular-compact" regular spaces in ZFC (you can guess the definition).