Strong convexity of the energy functional for negatively curved targets

Let $M$ be a Riemannian manifold and let $N$ be a Riemannian manifold of negative sectional curvature. Consider the energy functional on a connected component of $\mathcal{C}^{\infty}(M,N)$ that contains no map of rank everywhere at most $1$. Strong convexity conjecture. Strong convexity holds for the energy functional on this component. The preceding proposition establishes strict convexity under these hypotheses, but the authors state that no sufficient conditions for strong convexity are known in the smooth setting; they expect a quantitative refinement of the argument together with a Poincaré-type inequality to yield the claimed result.