Subadditivity of multiplier ideals with a pluriharmonic function

Question

I would like to have a reference for the following two facts (if true):

1. Let $D$ be a nef and big divisor on an algebraic variety $X$ and $h$ a Hermitian metric with minimal singularities on $D$, write $h=e^{-\phi}$ around some point $x\in X$ with $\phi$ being the local weight. Is it true that $\phi$ is a pluriharmonic function?

I am sure that the statement is true for ample divisors, and the fact that the multiplier ideals $\mathcal{J}(h^c)$ are trivial for every $c$ around $x$ gives me the feeling that this is true as well for $D$.

The second question is mostly motivated by the first:

2. Let $\mathcal{J}(\phi+\psi)$ be the multiplier ideal of the sum of a plurisubharmonic function and a pluriharmonic function, is it true that such ideal equals $\mathcal{J}(\phi)$?