The smooth Bogomolov–Miyaoka–Yau bound for exceptional circle orbits

Let $L$ be a 5-dimensional rational homology sphere with $H_1(L,\mathbb Z)=0$ admitting a fixed-point-free differentiable circle action. For an orbit $O$, define its exceptional multiplicity by

$$m(O):=\frac{|\operatorname{stab}(O)|}{\operatorname{lcm}\{|\operatorname{stab}(O_p)|:p\notin O\}}.$$

Let $O_1,\dots,O_k$ be the exceptional orbits. Generalized smooth circle-action Bogomolov–Miyaoka–Yau conjecture. Then

$$\sum_i\left(1-\frac{1}{m(O_i)}\right)\leq 3.$$

This generalizes the pseudo-free formulation to fixed-point-free actions that are not pseudo-free; the supplied source does not resolve it.