  This expository paper presents the current knowledge of particular fully
nonlinear curvature flows with local forcing term, so-called locally
constrained curvature flows. We focus on the spherical ambient space. The flows
are designed to preserve a quermassintegral and to de-/increase the other
quermassintegrals. The convergence of this flow to a round sphere would settle
the full set of quermassintegral inequalities for convex domains of the sphere,
but a full proof is still missing. Here we collect what is known and hope to
attract wide attention to this interesting problem.
