Johnson's determinantal identity for contiguous minors of Toeplitz matrices

Let $M_n(\mathbb{R})$ denote the real $n\times n$ matrices. For $A\in M_n(\mathbb{R})$, write $A_r(i,j)$ for the $r\times r$ contiguous submatrix beginning at row $i$ and column $j$. A matrix is Toeplitz if its entries are constant along diagonals, and let $J_n$ be the $n\times n$ matrix whose entries are all $1$.

Johnson's conjecture. For every $n\ge 2$ and every Toeplitz matrix $A\in M_n(\mathbb{R})$ satisfying $A+A^{\top}=2J_n$,

$$\ndet A_{n-1}(1,2)+\det A_{n-1}(2,1)=2\det A_{n-1}(1,1).$$

The identity is a determinantal relation among contiguous minors, motivated by Dodgson's condensation formula. The supplied source does not state whether Johnson's conjecture has been resolved.