A note on compatibility of special Hermitian structures

We prove that a compact Vaisman manifold $(M,J)$ cannot admit some type of special Hermitian metrics, such as special $k$-Gauduchon metrics, $p$-Kähler forms, Hermitian-symplectic or strongly Gauduchon metrics compatible to the same complex structure $J$. In particular, it cannot admit pluriclosed or balanced metrics. We also investigate the interplay between locally conformally symplectic forms taming the complex structure $J$ and special Hermitian structures.