Strongly Gauduchon conjecture for compact non-balanced BTP manifolds

Let $(M^n,g)$ be a compact Hermitian manifold satisfying the Bismut-parallel-torsion condition $\nabla^b T^b=0$, and suppose that $g$ is non-balanced. A Hermitian metric is strongly Gauduchon if there exists a global $(n,n-2)$-form $\Omega$ such that

$$\partial \omega^{n-1}=\overline{\partial}\Omega.$$

Strongly Gauduchon conjecture. The manifold $M$ does not admit any strongly Gauduchon metric. The claim extends the known obstruction for compact non-Kähler BKL manifolds; the source notes that it holds when $B\geq \operatorname{Ric}(Q)$, including the Vaisman case, but leaves the general statement as a conjecture.