Let $X=\{y=0\}\cup\{y=1\}\subset{\mathbb{C}}^{2}$ and $\eta\in{\mathcal{L}}(X)$, where

$$\eta(z)=\left\{\begin{array}[]{ll}\rho(1,z),\;\;\;\;\;\;\;{\rm if}\;z=(x,0),\\ \rho(1,z)+1,\;{\rm if}\;z=(x,1).\end{array}\right.$$

The function $\widetilde{\eta}$ is not $\omega$-psh on $\overline{X}=\{y=0\}\cup\{y=t\}$, hence $\eta$ does not have an extension in ${\mathcal{L}}({\mathbb{C}}^{2})$. Indeed, the maximum principle is violated along $\{y=0\}$ near the point $a=[0:1:0]$, since $\widetilde{\eta}([t:1:0])=0$ for $t\neq 0$, while $\widetilde{\eta}([t:1:t])=1$.