Projective bundles that admit coupled Kähler-Einstein metrics but no Kähler-Einstein metrics

Using Hultgren’s polytope formulation of the existence of coupled Kähler-Einstein (cKE) metrics on toric Fano manifolds, we construct explicit higher-dimensional toric Fano manifolds that admit two coupled Kähler-Einstein metrics but no ordinary Kähler-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 $n\ge 4$.