Jamison and Sprague's threshold-number conjecture for graphs and their complements

For a finite graph $G$, let $\Theta(G)$ be its threshold number, the smallest positive integer $k$ such that $G$ is a $k$-threshold graph, and let $G^c$ denote its complement.

Jamison and Sprague's conjecture. For every integer $k\ge 1$, there is a graph $G$ with

$$\Theta(G)=2k \quad\text{and}\quad \Theta(G^c)=2k+1.$$

Jamison and Sprague established that the threshold numbers of a graph and its complement differ by at most one, with the parity determining which value can occur. Chen and Hao gave a partial solution by determining the threshold numbers of complete multipartite graphs whose parts are not small and of their complements; the stated existence claim is resolved according to the supplied status evidence.