The $\mathcal{QHB}$ conjecture for small Seifert fibered $L$-spaces

Let $Y=Y(-2;r_1,r_2,r_3)$ be a small Seifert fibered space that is also an $L$-space. Let $\mathcal{QHB}$ denote the class of small Seifert fibered spaces admitting the relevant rational-homology-ball plumbing description, and let $\xi_{can}$ be the canonical contact structure.

The $\mathcal{QHB}$ conjecture. If $Y$ admits a symplectic rational homology ball filling, then

$$Y\in\mathcal{QHB}$$

and the filled contact structure is $\xi_{can}$.

The conjecture would show that, among the small Seifert fibered $L$-spaces with $e_0=-2$, every symplectic rational homology ball filling arises only in the classified $\mathcal{QHB}$ cases and uses the canonical contact structure. The source states that the authors do not know any example outside $\mathcal{QHB}$ admitting such a filling.