The canonical-connection Ambrose–Singer conjecture

Let $(M^n,g)$ be a compact Hermitian manifold. For real numbers $t$ and $s$, let $\nabla^{(t)}=(1-\frac{t}{2})\nabla+\frac{t}{2}\nabla^b$ be the $t$-Gauduchon connection, and define the canonical metric connection

$$D^t_s=(1-s)\nabla^{(t)}+s\nabla^g.$$

Here $\nabla^g$ is the Levi-Civita connection, and $(t,s)$ ranges over $\Omega=\{s\neq 1\}\cup\{(0,1)\}\subseteq\mathbb R^2$. The distinguished connections are $\nabla^b$, $\nabla^+=2\nabla^g-\nabla^b$, and $\nabla^-=2\nabla+\nabla^b-2\nabla^g$. Canonical-connection Ambrose–Singer conjecture. For any canonical connection $D^t_s$ other than $\nabla^b$, $\nabla^+$, and $\nabla^-$, if $D^t_s$ is Ambrose–Singer, then $g$ is Kähler. In particular, if such a $D^t_s$ is Ambrose–Singer and Ricci flat, then it is flat. This extends the proposed classification of Ambrose–Singer canonical connections beyond the Bismut, plus, and minus connections. The source presents it as a speculation, and no resolution is supplied.