Non-negative fat-tailed "almost stable" family of distribution with finite mean?

Question

I am looking for a finite-dimensional family of distributions $F_X(x)$ with all the following properties:

1. Supported on $[0, +\infty)$,
2. Fat tailed, i.e. $(1-F_X(x)) \sim x^{-\alpha}$ for $x\to +\infty$ and some $\alpha>0$,
3. The mean is finite (this implies $\alpha>1$ in 2.),
4. Stable under sum of i.i.d. variables: If $X_1,\ldots,X_n\sim F_X$ are i.i.d., then $Y=\sum_{j=1}^nX_j$ belongs to the same family?

There are several examples if any of the conditions is removed:

1. The symmetric stable distributions with $\alpha>1$.
2. Many examples here: Gamma, Inverse Gaussian just to cite two.
3. The Levy distribution (and other one-sided stable distributions with $0<\alpha<1$.
4. Many examples here too: Pareto, Burr, Log-Gamma, etc.

Is there a family satisfying all of them, or is there a reason why it is impossible?

Updated: Clarify that the family has to be finite-dimensional.