Gromov's long-neck conjecture for area non-increasing maps

Let $(M,g_M)$ be a compact $m$-dimensional Riemannian manifold with boundary and ${\rm scal}_M\geq m(m-1)$. Let $\Phi:M\to S^m$ be a smooth area non-increasing map that is locally constant near the boundary. Suppose

$${\rm dist}_{g_M}({\rm supp}({\rm d}\Phi),\partial M)\geq \frac{\pi}{m}.$$

Gromov's conjecture. Then ${\rm \deg}(\Phi)=0$.

This conjecture extends Llarull's sphere rigidity theorem to compact manifolds with boundary and is proposed as a statement about the obstruction created by a sufficiently long neck. Its resolution status is not specified in the source.