Stein-fillability conjecture for the edge contact structures on Brieskorn spheres

For $n>2$, consider the tight contact structures $\eta^n_{i,j}$ on $-\Sigma(2,3,6n-1)$ with $0\leq i\leq n-2$, $|j|\leq n-i-2$, and $j\equiv n-i\pmod{2}$, and the tight contact structures $\xi^n_{i,j}$ on $-\Sigma(2,3,6n+1)$ with $0\leq i\leq n-1$, $|j|\leq n-i-1$, and $j\not\equiv n-i\pmod{2}$. The edge Stein-fillability conjecture. The contact structures

$$\eta^n_{i,\pm(n-i-2)}\quad\text{are Stein fillable for }1\leq i\leq n-3,$$

and

$$\xi^n_{i,\pm(n-i-1)}\quad\text{are Stein fillable for }1\leq i\leq n-2.$$

The authors present this as a new conjecture after noting that the original non-Liouville-fillability conjecture is disproved by Stein fillings of $\eta^4_{1,\pm1}$. The inner-triangle cases are known to be non-Liouville fillable, while the asserted Stein fillability of all the edge cases remains open in the supplied text.