The Wigner-Yanase entropy is not subadditive

Wigner and Yanase introduced in 1963 the Wigner-Yanase entropy defined as minus the skew information of a state with respect to a conserved observable. They proved that the Wigner-Yanase entropy is a concave function in the state and conjectured that it is subadditive with respect to the aggregation of possibly interacting subsystems.
While this turned out to be true for the quantum-mechanical entropy, we negate the conjecture for the Wigner-Yanase entropy by providing a counter example.