  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
$n\leq 7$. Chu-Kwong-Lee showed in arXiv:2208.12240 [math.DG] a corresponding
rigidity statement when $n\leq 5$. In this paper, we show optimality of the
dimension constraints by giving concrete counterexamples in $n\geq 7$ and
extending the rigidity result to $n=6$. Concerning uniformly positive
intermediate curvature, we show that simply-connected manifolds with dimension
$\leq 5$ and bi-Ricci curvature $\geq 1$ have finite Urysohn 1-width.
Counterexamples are constructed in dimension $\geq 6$.
