Within a theoretical framework that accounts for all open-charm mesons, including the ${D}_{0}^{*}(2308)$, the ${D}_{sJ}^{*}(2317)$, and the ${D}_{sJ}(2460)$, we analyze the structure and explore possible quantum number assignments for the ${D}_{sJ}(2860)$ and the ${D}_{sJ}(2700)$ mesons reported by BABAR and Belle Collaborations. The open-charm sector is properly described if considered as a combination of conventional quark-antiquark states and four-quark components. All negative parity and ${2}^{+}$ states can be understood in terms only of $q\overline{q}$ components, however the description of the ${0}^{+}$ and ${1}^{+}$ mesons is improved whenever the mixing between two- and four-quark configurations is included. We analyze all possible quantum number assignments for the ${D}_{sJ}(2860)$ in terms of both $c\overline{s}$ and $cn\overline{s}\overline{n}$ configurations. We discuss the role played by the electromagnetic and strong decay widths as basic tools to distinguish among them. The broad structure reported by BABAR near 2.7 GeV is also analyzed.