Honda's classification conjecture for Moishezon twistor spaces on $4\bar{\mathbb{CP}}^2$

Let $Z$ be a Moishezon twistor space on $4\mathbb{CP}^2$, and let $k$ denote the parameter used to classify its fundamental cycle. A birational type space is one whose anticanonical map is birational, while a conic bundle type space is one whose anticanonical map has two-dimensional image and general fiber a rational curve. A Joyce metric is a self-dual metric in Joyce's class, and a minitwistor space is the quotient complex surface associated with a holomorphic $\mathbb{C}^*$-action on the twistor space.

Honda's classification conjecture. The following assertions should hold:

1. If $Z$ is of birational type with $k=6$, then $Z$ is a twistor space of a Joyce metric; in particular, it should admit an effective $\mathbb{C}^*\times\mathbb{C}^*$-action.
2. If $Z$ is of conic bundle type with $k=4$, then $Z$ is isomorphic to one of the twistor spaces constructed by Honda; in particular, it should admit a non-semifree $\mathbb{C}^*$-action.
3. If $Z$ is of conic bundle type with $k=5$, then it is a degenerate form of Honda's twistor spaces. In particular, it should admit a non-semifree $\mathbb{C}^*$-action, and its minitwistor space with respect to this action should be isomorphic to the corresponding Honda minitwistor space.

The conjecture proposes that the remaining birational and conic-bundle cases coincide with previously constructed families of highly symmetric twistor spaces. The source presents these identifications as expected rather than established, and the parser supplies no evidence of a resolution.