(Sketch) Given Proposition 3.32, one can set up a Banach iteration scheme to correct the volume form error. A subtlety caused by metric incompleteness is that the parametrix $P_{\nu}$ can only invert sources with compact supports. This problem can be circumvented using the extension norm trick as in Proposition 3.23, and we obtain a Calabi-Yau metric on a shrinked domain $M^{+}_{\nu-1}$. Changing $\nu$ to $\nu+1$ gives the statement. ∎