Small tree-foliation conjecture for positively curved Riemannian 3-spheres

Let $(S^3,g)$ be a Riemannian 3-sphere with scalar curvature $R_g\geq \Lambda_0>0$. A tree-foliation $\{\Sigma_t\}_{t\in T}$ is a foliation of $(S^3,g)$ by surfaces whose parameter space has the structure of a tree. For a leaf $\Sigma_t$, let

$$\operatorname{diam}(\Sigma_t)=\sup_{x,y\in\Sigma_t}\{\operatorname{dist}_{\Sigma_t}(x,y)\}$$

denote its intrinsic diameter. Small tree-foliation conjecture. There exists a constant $C>0$ such that $(S^3,g)$ admits a tree-foliation $\{\Sigma_t\}_{t\in T}$ satisfying, for every $t\in T$,

$$\operatorname{diam}(\Sigma_t)\leq \frac{C}{\sqrt{\Lambda_0}},\qquad \operatorname{Area}(\Sigma_t)\leq \frac{C}{\Lambda_0}.$$

Such a foliation would provide the small-area and small-intrinsic-diameter sphere foliation needed to construct a continuously varying sweepout by short closed curves. The statement is presented as a conjectural ingredient in the paper; no resolution is supplied here.