Projective bundles that admit coupled K\"ahler-Einstein metrics but no K\"ahler-Einstein metrics

Using Hultgren's polytope formulation of the existence of coupled K\"ahler-Einstein (cKE) metrics on toric Fano manifolds, we construct explicit higher-dimensional toric Fano manifolds that admit two coupled K\"ahler-Einstein metrics but no ordinary K\"ahler-Einstein metrics. In particular, we exhibit such examples among certain projective bundles over products of projective spaces. Motivated by these constructions, we conjecture that examples of this type exist in all dimensions $n\geq 4$.