  This paper aims to improve guaranteed error control for the Stokes problem
with a focus on pressure-robustness, i.e. for discretisations that compute a
discrete velocity that is independent of the exact pressure. A Prager--Synge
type result relates the velocity errors of divergence-free primal and perfectly
equilibrated dual mixed methods for the velocity stress. The first main result
of the paper is a framework with relaxed constraints on the primal and dual
method. This enables to use a recently developed mass conserving mixed stress
discretisation for the design of equilibrated fluxes and to obtain
pressure-independent guaranteed upper bounds for any pressure-robust (not
necessarily divergence-free) primal discretisation. The second main result is a
provably efficient local design of the equilibrated fluxes with comparably low
numerical costs. Numerical examples verify the theoretical findings and show
that efficiency indices of our novel guaranteed upper bounds are close to one.
