Let $\varepsilon\geq 0$ and $u$ be a continuous $(1+\varepsilon)\omega$-psh function on ${\mathbb{P}}^{n}$ so that $u(z)\leq 0$ for all $z\in{\mathbb{P}}^{n}$. If $c>1$ and $\varphi$ is an $\omega$-psh function on $X$ so that $\varphi<u$, then there exists a $c\omega$-psh function $\psi$ on ${\mathbb{P}}^{n}$ so that

$$\frac{1}{c}\,\psi(z)\leq\frac{1}{1+\varepsilon}\,u(z),\;\forall z\in{\mathbb{P}}^{n},$$

and

$$\psi(z)=\varphi(z)+(c-1)\theta(z)+(c-1)\min_{\zeta\in{\mathbb{P}}^{n}}u(\zeta),\;\forall z\in X.$$