Prove or disprove: $~\lim_{n\to \infty}\int |h_n - h|d\mu = 0$ when $h_n$ may not be non-negative.

Let $\{h_n\}$ be sequences of integrable functions on a measure space $(X,\mu)$. Suppose

(i) $h \in L^1(\mu).$

(ii) $\lim_{n\to \infty}h_n(x)=h(x)~$ for $\mu-$a.e. $x$.

(iii) $\lim_{n\to \infty}\int h_n d\mu = \int h d\mu.$

Prove or disprove: $~\lim_{n\to \infty}\int|h_n - h|d\mu = 0.$

Attempt 1:

Set $F_n = |h_n|-|h_n - h|$ so that $F_n \rightarrow |h|$ a.e., then $|F_n| \leq ||h_n|-|h_n-h|| \leq |h|$. We can apply Dominated Convergence Theorem to get $\lim_{n\to\infty}\int F_n d\mu = \int |h| d\mu.$ But

$$\lim_{n\to\infty}\int |h_n-h|d\mu = \lim_{n\to\infty}\int |h_n-h|-|h_n|+|h_n|=-\lim_{n\to\infty}\int F_n d\mu+\lim_{n\to\infty}|h_n|d\mu = -\int|h|d\mu+ \lim_{n\to\infty}\int |h_n|d\mu$$

I have no idea how to prove that the last equality equals to zero.

Attempt 2:

I also tried Fatou's Lemma: $$ 2\int h = \int \liminf_{n\to\infty} \left ( h_n + h - |h_n - h| \right ) \le \int h + \int h + \liminf_{n\to\infty} \left ( - \int |h_n - h| \right ). $$ If this holds, then the problem is solved. But here $-|h_n - h| \leq 0$, and Fatou's lemma is not applicable.

Any help would be appreciated.

Counter-example: On $(0,1)$ with Lebesgue measure let $h_n(x)=n(\chi_{(0,\frac 1 n)}-\chi_{(1-\frac 1 n,1)})$ and $h=0$. Here $\int h_n=\int h=0$ and $\int |h_n|=2$ for all $n$.