Classification conjecture for Lee-class sets of class VII surfaces

Let $S$ be a compact complex surface in the Kodaira class ${\rm VII}$. Write ${\mathcal T}(S)$ for its set of Lee classes, viewed in $H^1_{dR}(S,\mathbb R)$ and ordered along $(-\infty,0)$. Classification conjecture.

(a) If there exists a de Rham class $b\in(-\infty,0)\subset H^1_{dR}(S,\mathbb R)$ such that

$${\mathcal T}(S)\subset(-\infty,b),$$

then $S$ is obtained from either an Inoue--Bombieri surface or a hyperbolic Kato surface by blowing up points.

(b) If there exists a de Rham class $d\in(-\infty,0)\subset H^1_{dR}(S,\mathbb R)$ such that

$${\mathcal T}(S)\subset(d,0),$$

then $S$ is obtained from an Inoue--Bombieri surface by blowing up points.

The conjecture is motivated by the preceding classification results for Inoue--Bombieri, Hopf, hyperbolic Kato, and Enoki surfaces, together with Brunella's results on plurisubharmonic functions with analytic singularities. It proposes that sufficiently bounded Lee-class sets force the surface to belong to the corresponding known classes; the paper presents this as an expectation rather than an established classification.