The prescribed mean curvature infinitude conjecture

Let $(M^{n+1},g)$ be the ambient Riemannian manifold under consideration, let $h\in C^{\infty}(M)$, and let $H_{Y_p}$ denote the mean curvature of a hypersurface $Y_p\subset M$. Prescribed mean curvature infinitude conjecture. For any $h\in C^{\infty}(M)$, there exists infinitely many distinct hypersurfaces, $\{Y_p\}$, with

$$H_{Y_p}=h\big|_{Y_p}.$$

This proposes an analogue for constant or prescribed mean curvature of known infinitude results for embedded minimal hypersurfaces; the statement is presented as a hoped-for conjecture, and no resolution is supplied in the source.