  It has been conjectured by Fino and Vezzoni that a compact complex manifold
admitting both a compatible SKT and a compatible balanced metric also admits a
compatible K\"ahler metric. Using the shear construction and classification
results for two-step solvable SKT Lie algebras from our previous work, we prove
this conjecture for compact two-step solvmanifolds endowed with an invariant
complex structure which is either (a) of pure type or (b) of dimension six. In
contrast, we provide two counterexamples for a natural generalisation of this
conjecture in the homogeneous invariant setting. As part of the work, we obtain
further classification results for invariant SKT, balanced and K\"ahler
structures on two-step solvable Lie groups. In particular, we give the full
classification of left-invariant SKT structures on two-step solvable Lie groups
in dimension six.
