Further reduction of the torus rank

Determine whether the torus-rank assumption in the even-dimensional positive second intermediate Ricci curvature Euler-characteristic result can be reduced to rank six or rank five.

Background

After proposing the T^7 Euler-characteristic positivity conjecture, the paper explicitly identifies a further unresolved issue: whether the required torus symmetry can be reduced even more. The question is motivated by the known example with even dimension, positive second intermediate Ricci curvature, Euler characteristic zero, and torus symmetry of rank four.

The unresolved range is specifically whether the relevant torus rank can be lowered from seven to six or five; the paper does not resolve either possibility.