Rigidity conjecture for maximal-rank torus actions on rationally elliptic manifolds

Let $M^n$, with $n\geq 3$, be a smooth, closed, simply connected, rationally elliptic $n$-manifold equipped with a smooth, effective action of the torus $T^k$ of rank

$$k=\left\lfloor\frac{2n}{3}\right\rfloor.$$

Rigidity conjecture. $M^n$ is equivariantly diffeomorphic to an effective, linear action of $T^k$ on a manifold of one of the following forms:

$$X\times\prod \mathrm{\mathbb{S}}^3,$$

where $X\in\{\mathrm{\mathbb{S}}^3,\mathrm{\mathbb{S}}^4,\mathrm{\mathbb{S}}^5,\mathrm{\mathbb{S}}^5\times\mathrm{\mathbb{S}}^5,\mathrm{\mathbb{S}}^7\}$;

$$(Y\times\prod \mathrm{\mathbb{S}}^3)/\mathrm{\mathbb{S}}^1,$$

where $Y\in\{\mathrm{\mathbb{S}}^3,\mathrm{\mathbb{S}}^5\}$; or

$$(\prod \mathrm{\mathbb{S}}^3)/T^2.$$

The theorem preceding this conjecture establishes the corresponding classification up to rational homotopy type, while the conjecture asks for equivariant diffeomorphism. Partial results are known in low dimensions, but the full classification remains open.