Lemma 3.7 is incorrect. This gap has the following consequences:
Theorems A, 3.1, 3.2, and 3.3 are incorrect.
To my knowledge, the statement of Theorem A is only known to be true when M = SU(2). For a generalization to the case where M is any compact semisimple Lie group, see Theorem E in my paper with Miguel Domínguez-Vázquez and David González-Álvaro.
Remark 3.5 is incorrect, unless M = N = SU(2).
Many of the consequences presented in Section 3.3 do not hold.
The Cheeger deformation of the product metric on SU(2) x SU(2) = S^3 x S^3 by the diagonal SU(2) action by left multiplication results in a family of metrics that have Ric2 > 0. These metrics are invariant under the diagonal SU(2) action by left multiplication and the action of SU(2) x SU(2) by right multiplication. By taking appropriate quotients by circle subactions, S^2 x S^3 and S^3 x S^2 each admit metrics with Ric2 > 0 and symmetry rank 2, and S^2 x S^2 admits a metric with Ric2 > 0 and symmetry rank 1. Therefore, some of the consequences in Section 3.3 do hold in low dimensions.
In Example 5.5, only S^3 x S^3 is known to admit a metric with Ric2 > 0 that is invariant under a free T^2 action.