  The class of $L\Sigma(\leq\omega)$-spaces was introduced in 2006 by Kubi\'s,
Okunev and Szeptycki as a natural refinement of the classical and important
notion of Lindel\"of $\Sigma$-spaces. Compact $L\Sigma(\leq\omega)$-spaces were
considered earlier, under different names, in the works of Tkachuk and
Tkachenko in relation to metrizably fibered compacta. In this paper we give
counterexamples to several open questions about compact
$L\Sigma(\leq\omega)$-spaces that are scattered in the literature. Among other
things, we refute a conjecture of Kubi\'s, Okunev and Szeptycki by constructing
a separable Rosenthal compactum which is not an $L\Sigma(\leq\omega)$-space. We
also give insight to the structure of first-countable
$(K)L\Sigma(\leq\omega)$-compacta.
