The smooth Bogomolov–Miyaoka–Yau bound for pseudo-free circle actions

Let $L$ be a 5-dimensional rational homology sphere with $H_1(L,\mathbb Z)=0$ admitting a pseudo-free differentiable circle action. Let $O_1,\dots,O_k$ be the nonfree orbits with stabilizers $\mathbb Z/m_1,\dots,\mathbb Z/m_k$. Smooth circle-action Bogomolov–Miyaoka–Yau conjecture. Then

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

This follows formally from the proposed smooth topological Bogomolov–Miyaoka–Yau inequality via the quotient 4-manifold, but the source does not establish either conjecture.