Chern-Ricci flat balanced metrics on small resolutions of Calabi-Yau threefolds

Given a (smoothable) projective nodal Kähler Calabi-Yau threefold, we show, via a gluing construction, that all its - possibly non-Kähler - small resolutions admit Chern-Ricci flat balanced metrics, which among other things solve the dilatino equation appearing in the Hull-Strominger system.