TITLE: Regularity properties of Turing-invariant and arbitrary sets of reals
QUESTION [6 upvotes]: The question whether Turing determinacy implies $AD$ is a well-known open problem. I was wondering if anything is known about the following analogous question:
Let $\Gamma$ be a regularity property (e.g. Lebesgue measurability, Baire property, $\mathbb P$-measurability for a reasonable arboreal forcing $\mathbb P$, etc). Assuming $ZF+DC+"\Gamma$ holds for all Turing-invariant sets of reals$"$, does it follow that $\Gamma$ holds for all sets of reals?

REPLY [2 votes]: Here is not an answer but an example:
Let PSP be the statement that every uncountable set of reals has a perfect subset and TPSP that every uncountable set of Turing degrees has a perfect subset.
Then PSP and TPSP are equivalently consistent over $ZF+DC$. 
Clearly over $ZF+DC$, PSP implies TPSP but I don't know whether the inverse is true.