K\"ahler-Einstein toric submanifolds of the projective space

We show that the K\"ahler-Einstein metrics on the four families of examples of symmetric toric Fano manifolds presented by Batyrev and Selivanova cannot be realized as metrics induced by immersions into projective spaces equipped with Fubini-Study metrics. We obtain a similar conclusion for the non-symmetric examples discovered by Nill and Paffenholz. A consequence is that a centrally symmetric toric Fano manifold admits a K\"ahler-Einstein metric induced by a projective immersion if and only if it is a product of projective lines. These results provide evidence for a broader conjecture characterizing which K\"ahler-Einstein metrics can be induced by projective immersions.