A counterexample to the conjecture on monotonicity of an integral with respect to Gaussian measure

It is shown that for the Kantorovich metrics $\varkappa$ on probability measures for centered Gaussian measures $\gamma$ defined on Euclidean space $E$ of random variables $X$ the integral
$$ I(\gamma)=\iint\limits_{E\oplus E}\varkappa(\mathscr L(X_1),\mathscr L(X_2))(\gamma\otimes\gamma)\,d(X_1,X_2), $$
is not always monotonic in $\gamma$.