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.