The determinant-positive extension of the non-Eguchi–Hanson obstruction theorem

Let $L_\infty$ be the limiting linear map appearing in the desingularization and obstruction setup, where the theorem was originally stated under the assumption $L_\infty=\operatorname{Id}$. Determinant-positive extension conjecture. The statement of the non-Eguchi–Hanson obstruction theorem remains valid without assuming $L_\infty=\operatorname{Id}$, provided only that

$$\det(L_\infty)>0.$$

This is stated as a belief immediately after a remark explaining that the result likely holds for general $L_\infty$, subject to control of the compactness of the lattice it generates. The source gives no proof or resolution of this extension.