Spherical connectedness for profiles with at least two degree-one faces

Prove that, for every spherical profile $(\mu,k)$ with no restrictions on the number $\mu_1$ of degree-one faces, the set of square-tiled surfaces $ST(\mu,k)$ is connected by cylinder shears.

Background

The paper proves spherical connectedness under ordinary cylinder shears when $\mu_1\le 1$, while the general cylinder-shear conjecture would also require profiles with $\mu_1\ge 2$. The authors formulate this as a separate conjecture extending their spherical result.

This problem concerns regular cylinder shears only; half-cylinder shears allow broader spherical results in the paper but are not part of the conjecture stated here.