Dimension constraints in some problems involving intermediate curvature

In arXiv:2207.08617 [math.DG] Brendle-Hirsch-Johne proved that $T^m\times S^{n-m}$ does not admit metrics with positive $m$-intermediate curvature when $n\le 7$. Chu-Kwong-Lee showed in arXiv:2208.12240 [math.DG] a corresponding rigidity statement when $n\le 5$. In this paper, we show the sharpness of the dimension constraints by giving concrete counterexamples in $n\ge 7$ and extending the rigidity result to $n=6$. Concerning uniformly positive intermediate curvature, we show that simply-connected manifolds with dimension $\le 5$ and bi-Ricci curvature $\ge 1$ have finite Urysohn 1-width. Counterexamples are constructed in dimension $\ge 6$.