  Yang-Baxter deformations of superstring sigma-models have recently inspired a
supergravity solution generating technique. Using the open/closed string map
and a Killing bi-vector as a deformation parameter, new solutions can be built,
such that the (generalised) supergravity field equations were conjectured to
always reduce to the classical Yang-Baxter equation (CYBE). In this work we
provide a proof of this conjecture, using a systematic approach based on the
so-called $\beta$-supergravity, which is a dynamical theory for the field
$\beta^{mn}$ instead of the NSNS 2-form $b_{mn}$.
