Let $X_1, X_2, \dots, X_n$ be jointly Gaussian random variables with covariance $\Sigma \in \mathbb{R}^{n \times n}$. The mean does not really matter for this question and we can assume it to be zero for convenience. Also, $\text{Var}(X_i) = 1$ for all $i = \{1,2,\dots, n\}$.
Given $\mathcal{S} \subseteq \{1,2, \dots, n\}$, define $\sigma^{2}(X_i; \mathcal{S}) = \text{Var}(X_i|\{ X_j : j \in \mathcal{S} \})$, i.e., the conditional variance of $X_i$ given $X_j$'s corresponding to the indices in the set $\mathcal{S}$. Lastly, let $\pi$ be a permutation of $\{1,2,\dots, n\}$ and $\pi(i)$ denote the $i^{\text{th}}$ element of the permutation. We define the following function:
$$ f(\pi) = \sum_{i = 1}^{n} \sigma(X_{\pi(i)}; \pi_{[1:(i-1)]}),$$ where $\pi_{[1:k]}$ denotes the set corresponding to the first $k$ elements of the permutation $\pi$.
I am interested in the range of the function $f$ over all possible permutations $\pi$. Intuitively speaking, I would expect this range to be small and obviously dependent on $\Sigma$. I have tried several simulations and the results have exhibited this intuitive behaviour. However, I have been having a difficult time to characterise or even bound this range.
I am even happy to be able to prove the following relation: $f(\pi_1) \leq a f(\pi_2) + b$ for any two permutations, $\pi_1$ and $\pi_2$ and constants $a, b > 0$ (apart from the trivial solutions for $a,b$). If it helps, you can also restrict only to cyclic permutations of the set $\{1,2,\dots, n\}$.
Any leads or references will be highly appreciated. Thanks!