Conjecture on rank-one solutions for edge sums in the max-cut SDP

Let $G_1$ and $G_2$ be two graphs with vertex sets

$$V_1=\{1,2,\ldots,m+1,m+2\},\qquad V_2=\{m+1,m+2,\ldots,n-1,n\}.$$

Let $(X_1,S_1)$ and $(X_2,S_2)$ be the primal-dual solution pairs to the max-cut SDPs on $G_1$ and $G_2$, respectively. Let $G$ be the edge sum of $G_1$ and $G_2$, with common edge $(m+1,m+2)$, and let $(X^*,S^*)$ be the primal and dual optimal solutions for $G$. Edge-sum conjecture. The following statements are equivalent: (1) $X_1$ and $X_2$ are rank-one matrices and agree on the intersection; (2) for some common choice of signs on the off-diagonal $w_1$ entries, the dual solution has the form

$$S=\begin{bmatrix} S_1^* & 0 \\ 0 & 0 \end{bmatrix}+\begin{bmatrix} 0 & 0 \\ 0 & S_2^* \end{bmatrix}-\frac14\begin{bmatrix} 0&0&0&0\\0&w_1&\pm w_1&0\\0&\pm w_1&w_1&0\\0&0&0&0\end{bmatrix}.$$

The first two matrices are $n\times n$; $S_1$ is $(m+2)\times(m+2)$ and $S_2$ is $(n-m)\times(n-m)$, while in the final matrix the $w_1$ entries occur in positions $(m+1,m+1)$ and $(m+2,m+2)$ (with the off-diagonal entries carrying the common sign). This conjecture concerns how rank-one primal solutions and the corresponding dual structure behave under an edge sum, extending the paper's results for vertex sums and special edge sums. Its resolution for general edge sums or larger clique sums is left as future work.