Let $M$ be a closed manifold with non-torsion $\pi_2$, and $A$ a non-trivial free homotopy class of a map $f: S^2 \to M$. Let $S$ be the set of (immersed) class $A$ surfaces in $(M,g)$ with mean curvature bounded from above by a fixed constant $C$. Is there a $C=C(A)>0$ sufficiently small, so that the $g$-area function is bounded on $S=S(A,C)$?

A preliminary version of the question would be to ask if this holds for loops in $M$, with mean curvature replaced by geodesic curvature, again assuming $\pi_1(M)$ is non-torsion. But it is surfaces I need.