Conjecture that Gauduchon hyperbolicity implies balanced hyperbolicity

Let $X$ be a compact balanced manifold. A balanced metric $\omega$ is Gauduchon hyperbolic when it has the Gauduchon hyperbolicity property, and $X$ is balanced hyperbolic when it has the corresponding balanced hyperbolicity property.

Gauduchon-to-balanced hyperbolicity conjecture. If $X$ admits a balanced metric $\omega$ which is Gauduchon hyperbolic, then $X$ is balanced hyperbolic.

The conjecture proposes that Gauduchon hyperbolicity is strong enough to imply balanced hyperbolicity. It is presented as being in the spirit of a conjecture from Khelifati, but no resolution is supplied in the source.