Maximal symmetry rank conjecture for non-negatively curved manifolds

Let $T^k$ act isometrically and effectively on $M^n$, where $M^n$ is a closed, simply connected, non-negatively curved Riemannian manifold. Define

$$\mathcal{Z}=\prod_{i\leq r} S^{2n_i-1}\times\prod_{i>r}S^{2n_i},$$

where $n_i\geq 2$, $r=2\lfloor 2n/3\rfloor-n$, $0\leq m\leq 2n\mod 3$, and the $T^m$-action on $\mathcal{Z}$ is free and linear. Maximal symmetry rank conjecture. One has $k\leq\lfloor 2n/3\rfloor$. If $k=\lfloor 2n/3\rfloor$, then $M^n$ is equivariantly diffeomorphic to $\mathcal{Z}/T^m$ with a linear $T^k$-action.

This proposes both an upper bound for the symmetry rank and a classification in the equality case. The surrounding text states that maximal symmetry rank has not yet been established in all dimensions, so the conjecture remains open.