For a symmetric $p\times p$ matrix $X$, where $X$ has sparsity $E$ ($X_{ij} = 0$ if $i\neq j$ and $\{i,j\}\not\in E$) we say that $X$ has a positive semidefinite completion if there exists some $Z$ positive semidefinite and $X_{ij} = Z_{ij}$ whenever $i = j$ or $\{i,j\}\in E$.
If $E$ is a chordal sparsity pattern, there exist efficient ways of computing the maximum determinant completion $Z_c$, largely because the inverse of $Z_c$ is positive semidefinite with sparsity $E$ (so we can represent $Z_c$ implicitly with $Z_c^{-1}$).
My question is, is this the only reason why the maximum determinant completion is important (and heavily studied), or are there other practical benefits to this particular completion? I ask this because it seems like in general $Z_c$ is not a low-rank completion (the fact that $Z_c^{-1}$ exists implies $Z_c$ is full rank), which seems to me more practically beneficial, and almost equally easy to compute efficiently.
Can anyone shed some light on the historical and current uses of the maximum determinant completion, outside of this relationship with chordal sparse positive semidefinite matrices? Thanks!