The constant holomorphic sectional curvature conjecture for t-Gauduchon connections

Let $(M^n,g)$ be a compact Hermitian manifold of complex dimension $n\geq 2$. For a real number $t$, let $\nabla^{(t)}=(1-t)\nabla+t\nabla^b$ be the $t$-Gauduchon connection, let $R^{(t)}$ be its curvature, and let $H^{(t)}$ be its holomorphic sectional curvature. Constant $t$-Gauduchon holomorphic sectional curvature conjecture. If

$$t\neq 0,1,-1$$

and $H^{(t)}=c$ is constant, then $g$ must be Kähler. This generalizes the constant-curvature question for the Chern connection; the supplied text gives no resolution of the conjecture.