Preserving positive intermediate curvature

Consider a compact manifold $N$ (with or without boundary) of dimension $n$. Positive $m$-intermediate curvature interpolates between positive Ricci curvature ($m=1$) and positive scalar curvature ($m=n-1$), and it is obstructed on partial tori $N^n=M^{n-m}\times {\mathbb{T}}^m$. Given Riemannian metrics $g,\bar{g}$ on $(N,\mathrm{\partial }N)$ with positive $m$-intermediate curvature and $m$-positive difference $h_g-h_{\bar{g}}$ of second fundamental forms we show that there exists a smooth family of Riemannian metrics with positive $m$-intermediate curvature interpolating between $g$ and $\bar{g}$. Moreover, we apply this result to prove a non-existence result for partial torical bands with positive $m$-intermediate curvature and strictly $m$-convex boundaries.