So you want $$\forall x,\\, x^t (\Lambda^{-1} - \Sigma^{-1}) x \geq 0$$
$(\Lambda^{-1} - \Sigma^{-1})$ is a symmetric matrix, and the condition expresses that it must be a positive matrix too. Let $T = \Sigma^{-1}$ and $M = \Lambda^{-1}$ we want to maximize $\det(T)$ with the constraint $\det(T-M)\geq 0$

Let's write the KKT conditions

$$L(T,\mu_0,\mu_1) = -(\det(T)) +\mu_0 \det(T-M) +\mu_1 \det(T)$$

$$\left\\{\begin{array}{ccc}
\forall i,\\,-\Pi_{j\neq i} t_i + \mu_0 \hbox{adj}(T-M)_{i,i} + \mu_1 \Pi_{j\neq i} t_i &=  &0\\\\
\mu_0 & \leq & 0\\\\
\mu_1 & \leq & 0\\\\
\mu_0 \det(T-M) & = & 0\\\\
\mu_1 \det(T) & = & 0\\\\
\end{array}\right.$$

The last equation implies $\mu_1 = 0$ since we can always do better than an infinite volume. The other one tells us that the ellipses touch, so $\det{T-M}=0$
$$\forall i,\\, \Pi_{j\neq i} t_i = \mu_0 \hbox{adj}(T-M)_{i,i}$$

I don't see immediately an exact way to solve this system of equations.