Suppose we have a non trivial free second homotopy class $A$ of $M$. Assume $\pi_2(M)$ is nontorsion for $M$ a closed manifold. 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 the $g$-area function bounded on $S$?

I 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.