Dense subalgebras of von Neumann algebras and increasing nets

Let $N$ be a von Neumann algebra, and $A$ be a dense $*$-subalgebra of $N$ (in the ultraweak topology) with $A''=N$. Is it true that:

For any $x\in N^+$, there exists a increasing net $(x_j)$ in $A^+$ such that $x_j \to x$ in the ultraweak topology ?

The case of $A$ being an ideal of $N$ seems known (it is right?) but my question is about the general case.

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The following might be a counterexample. Let $N=L^{\infty}([0,1])$ and $A=C([0,1])$, the subalgebra of continuous functions. Take $x\in N\setminus \{0\}$ to be a nonnegative function such that the set $\{t\in [0,1]:x(t)=0\}$ has positive measure in every subinterval. The only element $f\in A^+$ such that $f\le x$ is $f\equiv 0$.