Cardinal invariants of cellular-Lindelof spaces

A space $X$ is said to be “cellular-Lindel"of” if for every cellular family $\mathcal{U}$ there is a Lindel"of subspace $L$ of $X$ which meets every element of $\mathcal{U}$. Cellular-Lindel"of spaces generalize both Lindel"of spaces and spaces with the countable chain condition. Solving questions of Xuan and Song, we prove that every cellular-Lindel"of monotonically normal space is Lindel"of and that every cellular-Lindel"of space with a regular $G_{\delta }$-diagonal has cardinality at most $2^{\mathfrak{c}}$. We also prove that every normal cellular-Lindel"of first-countable space has cardinality at most continuum under $2^{<\mathfrak{c}}=\mathfrak{c}$ and that every normal cellular Lindel"of space with a $G_{\delta }$-diagonal of rank $2$ has cardinality at most continuum.