Denote the standard Gaussian probability measure on $\mathbb R^n$ by $\gamma$. We partition $\mathbb R^n$ into two sets $A$ and $A^c$ such that $\gamma(A) = \gamma(A^c) = 1/2$.

Denote by $\gamma_{A}$ to Gaussian measure restricted to $A$, and normalized so that it is a probability measure. Similarly, define $\gamma_{A^c}$ to be the Gaussian measure restricted to $A^c$ and normalized.

My question is the following:

What is the optimal $A$ such that $\gamma_A$ and $\gamma_{A^c}$ are the farthest apart; i.e., solving

$$\arg\max_{A} W_2(\gamma_A, \gamma_{A^c}),$$

where $W_2$ is the 2-Wasserstein distance?

Possible generalization: Instead of constructing $\gamma_A$ and $\gamma_{A^c}$ as above, we could start with any two probability measures $\gamma_1$ and $\gamma_2$ such that $\gamma = \frac{\gamma_1 + \gamma_2}{2}$ and find $\arg \max_{\gamma_1, \gamma_2} W_2(\gamma_1, \gamma_2)$.

Finding upper bounds on the $W_2$ distance is also of interest. A natural conjecture, inspired by the Gaussian isoperimetric inequality, would be that $A$ should be a half-space. Counterexamples to this are also welcome!