Some notation: $A(\alpha|\beta)$ is the submatrix of $A \in \mathbb{R}^{n \times n}$ with with rows $\alpha$ and columns $\beta$. $\textrm{det } A(\alpha|\alpha) =: \textrm{det } A(\alpha)$ are the principal minors of $A$. We define $\textrm{det } A(\emptyset)=1$. A matrix $A$ is irreducible if there is no permutation matrix $P$ such that

$$ P^{-1} A P = \begin{bmatrix} E & G \\ 0 & F \end{bmatrix} $$

where $E \in \mathbb{R}^{k \times k}$ and $F \in \mathbb{R}^{n-k \times n-k}$ are square matrices for $k \geq 1$.

Conjecture: Given an invertible, irreducible matrix $A \in \mathbb{R}^{n \times n}$ with $\textrm{det } A(\alpha) = \textrm{det } (({A}^{-1})(\alpha))$ for all $\alpha \subseteq \{1,\dots,n\}$, there exists a non-singular diagonal matrix $D$ with $D^{-1} A D = A^{-1}$ or $D^{-1} A^T D = A^{-1}$.

This has been shown to be true for $n = 2$ and $n=3$ (shown in "On matrices having equal corresponding principal minors" by D.J. Hartfiel and R. Loewy). For $n \geq 4$, there is an additional requirement on $A$ that for any partition $\alpha, \beta$ of $\{1,\dots,n\}$ with $|\alpha|\geq 2, |\beta|\geq 2$ either $\textrm{ rank } A(\alpha|\beta) \geq 2$ or $\textrm{ rank } A(\beta|\alpha) \geq 2$ (shown in "Principal minors and diagonal similarity of matrices" by R. Loewy).

However, this example with $A^T = A^{-1}$ shows that the rank condition is not necessary:

$$A=\frac15\left( \begin{array}{cccc} -1 & 4 & -2 & -2 \\ -4 & 1 & 2 & 2 \\ 2 & 2 & -1 & 4 \\ -2 & -2 & -4 & 1 \\ \end{array} \right)$$