The HL-clan-reversal conjecture for skew-symmetric matrices

Let $A$ and $B$ be $n\times n$ skew-symmetric real matrices. Two such matrices have equal corresponding principal minors of all orders if, for every subset $X\subseteq [n]$, the principal minors of $A[X]$ and $B[X]$ are equal. They are HL-clan-reversal-equivalent when there is a sequence of $n\times n$ skew-symmetric matrices $A_0=A,\ldots,A_m=B$ such that, for each $k=0,\ldots,m-1$, $A_{k+1}=\operatorname{Inv}(X_k,A_k)$ for an HL-clan $X_k$ of $A_k$; here, for a skew-symmetric matrix $C=[c_{ij}]$ and $X\subseteq[n]$, $\operatorname{Inv}(X,C)=[t_{ij}]$ is defined by

t_{ij}=\begin{cases}-c_{ij},&i,j\in X,\c_{ij},&\text{otherwise},\end{cases}

and $X$ is an HL-clan of $C$ when both $C[X,\overline X]$ and $C[\overline X,X]$ have rank at most $1$, with $\overline X=[n]\setminus X$. HL-clan-reversal conjecture. Two $n\times n$ skew-symmetric real matrices have equal corresponding principal minors of all orders if and only if they are HL-clan-reversal-equivalent. The construction by HL-clan reversals preserves corresponding principal minors, and the conjecture proposes the converse characterization; its resolution is not specified in the source.