Spherical connectedness conjecture for square-tiled surfaces

Let $(\mu,k)$ be a spherical profile, meaning profile data for square-tiled surfaces on the sphere. Let $\operatorname{ST}(\mu,k)$ denote the corresponding set of square-tiled surfaces, and let cylinder shears be the allowed reconfiguration moves. Spherical connectedness conjecture. The set

$$\operatorname{ST}(\mu,k)$$

is connected by cylinder shears. This is a special case of the general connected-components conjecture and extends the known spherical connectedness theorem to spherical profiles with $\mu_1\ge 2$. The paper proves the spherical result in several cases, including all strata when half-shears are allowed, but the stated cylinder-shear conjecture remains open.