Uncentered Blaschke-Santaló inequalities for the Gaussian measure

We study the maximizers of the generalized volume product \[ γ_σ^n(A)\,γ_σ^n(A^\circ) \] among all measurable subsets $A\subset\mathbb{R}^n$, where $A^\circ$ denotes the polar set of $A$, and where $γ_σ^n$ denotes the centered Gaussian probability measure on $\mathbb{R}^n$ with covariance $σ^2 I_n$,