Let $\Omega$ be an open set in $\mathbb C^n$, and let $A$ be a closed pluripolar set in $\Omega$. Is there a notion of dimension of $A$ such that the following theorem is true?

Theorem.

Let $\phi$ be a plurisubharmonic function on $\Omega \setminus A$ (not necessarly assumed to be locally bounded above near $A$), and assume that the (real) codimension of $A$ is at least $3$. Then the function $\phi$ extends to a plurisubharmonic function on $\Omega$.

I think I can prove the theorem in case $A$ is complex analytic.