Sufficient Condition for Uniform Integrability $\mathcal{L}^1$-boundedness

Question

I saw the following proof showing necessary and sufficient conditions for Uniform Integrability:

And was just wondering does uniform integrability still hold if we ONLY have $\mathcal{L}^1$-boundedness and not the condition 2?

I.e. Is $\mathcal{L}^1$-boundedness alone for the family $\mathcal{X}$ of measurable functions sufficient for Uniform Integrability of $\mathcal{X}$?

If so - how could you go about showing it (cause I've been stuck on trying to show this)...or more than likely...

If not, why does this "intuitively" not work and could you please give a counter example?

Thanks in advanced!