  We present an hp-adaptive virtual element method (VEM) based on the
hypercircle method of Prager and Synge for the approximation of solutions to
diffusion problems. We introduce a reliable and efficient a posteriori error
estimator, which is computed by solving an auxiliary global mixed problem. We
show that the mixed VEM satisfies a discrete inf-sup condition, with inf-sup
constant independent of the discretization parameters. Furthermore, we
construct a stabilization for the mixed VEM, with explicit bounds in terms of
the local degree of accuracy of the method. The theoretical results are
supported by several numerical experiments, including a comparison with the
residual a posteriori error estimator. The numerics exhibit the p-robustness of
the proposed error estimator. In addition, we provide a first step towards the
localized flux reconstruction in the virtual element framework, which leads to
an additional reliable a posteriori error estimator that is computed by solving
local (cheap-to-solve and parallelizable) mixed problems. We provide
theoretical and numerical evidence that the proposed local error estimator
suffers from a lack of efficiency.
