The choice of perturbing the gravitational instanton $N_{i}$ of type $A_{k_{i}-1}$ by adding a linear function on $\mathbb{R}^{3}$ can be regarded as an intermediate choice between resolving the singularity of $M^{\textup{gh}}_{\epsilon}$ by assuming all punctures $p_{i}$ have weight $k_{i}=1$ and the direct gluing of $N_{i}$ to $M^{\textup{gh}}_{\epsilon}$. While not strictly necessary, the choice of perturbing $N_{i}$ by a linear function makes the exposition more uniform. In particular, the closed definite triple we will construct below fails to be hyperkähler by the same amount in a neighbourhood of $q_{j}$ and $\pm p_{i}$.