Adjacency-spectral determination conjecture for complements of multicone cycle graphs

Let $K_w$ be the complete graph on $w$ vertices, let $C_n$ be the cycle graph on $n$ vertices, and let $mC_n$ denote the disjoint union of $m$ copies of $C_n$. Write $G\bigtriangledown H$ for the join of graphs $G$ and $H$, and let $\overline{G}$ denote the complement of $G$. A graph is DS with respect to its adjacency spectrum if every graph with the same adjacency spectrum is isomorphic to it. Complementary multicone conjecture. The graphs $\overline{K_w\bigtriangledown mC_n}$ are DS with respect to their adjacency spectra. The source establishes this for some cases, including $\overline{K_w\bigtriangledown mC_3}$ and $\overline{K_w\bigtriangledown C_4}$, while the general claim remains unresolved in the source.