Question:DoesCon($ZF$)implyCon($ZF$ + $DC$ + "there is no paradoxical Banach-Tarski decomposition of the unit ball")?

Here **Con**($X$) is the consistency of $X$; $DC$ is dependent choice.

**Motivation for the Question**: Since the "paradoxical" sets in the Banach-Tarski theorem have to be non-measurable, Solovay's model of "every subset of reals is measurable" shows:

**Theorem**. **Con**($ZF$+ **Inacc**) *implies* **Con**($ZF$ + $DC$ + "*there is no paradoxical Banach-Tarski decomposition of the unit ball*").

In the above **Inacc** is the statement "there is an inaccessible cardinal".

Note that Shelah's model [Israel Journal of Math, 1984], constructed only
from **Con**($ZF$), in which $DC$ holds and all sets of reals have the Baire
property, is of no help in answering the above question since Dougherty and
Foreman [ J. Amer. Math. Soc., 1994] have shown that there are paradoxical
decompositions of the unit ball using pieces which *have the property of
Baire*. I do not know how much choice is needed in their construction.

This question was posed a while ago on an FOM-posting of mine, but remained unanswered.