In Besse's "EInstein manifolds", p. 354, the question is posed if the volume of Einstein metrics on a given compact manifold (normalized such that $Ric=\pm(n-1)g$) take only finitely many values.

For negative values, he says that no example of more than one volume on a fixed manifold is known. Are there still no such examples?

I'd like to know if there is any recent progress which is related to this question.

In Besse's "EInstein manifolds", p. 354, the question is posed if the volume of Einstein metrics on a given compact manifold (normalized such that $Ric=\pm(n-1)g$) take only finitely many values.

For negative values, they say that no example of more than one volume on a fixed manifold is known. Are there still no such examples?

I'd like to know if there is any recent progress which is related to this question.