Non-Stein-fillability conjecture for the contact structures on $-\Sigma(2,3,6n-1)$

Let $n\geq 2$, and let $\eta_{i,j}^n$ denote the contact structures on $-\Sigma(2,3,6n-1)$ indexed by

$$\mathcal P_n=\left\{(i,j)\in\mathbb Z\times\mathbb Z:0\leq i\leq n-2,\ |j|\leq n-i-2\text{ with }j\equiv n-i\pmod 2\right\}.$$

A contact structure is Stein fillable if it occurs as the boundary contact structure of a Stein filling.

Non-Stein-fillability conjecture. The contact structures $\eta_{i,j}^n$ are not Stein fillable if $i>0$.

The contact structures in the bottom row, with $i=0$, are Stein fillable, while all the other contact structures are known to be strongly symplectically fillable. The top contact structure is known not to be Stein fillable, but no Stein filling is known for the remaining cases with $i>0$.