The injectivity radius--spherical radius comparison conjecture

Let $(M^n,g)$ be a complete Riemannian manifold. Write $\operatorname{Inj}(M)$ for its injectivity radius and $\operatorname{Rad}_{\mathbb S^n}(M)$ for its spherical radius. The injectivity radius--spherical radius comparison conjecture. There exists a universal constant $c_n$ such that

$$\operatorname{Inj}(M)\leq c_n\operatorname{Rad}_{\mathbb S^n}(M).$$

The conjecture is motivated by the inequalities relating injectivity radius, filling radius, Uryson width, and spherical radius. The source gives no resolution; it would turn bounds on spherical radius from positive scalar curvature into bounds on injectivity radius.