The central Delannoy formula for subdivisions of a 2 × n grid

Let an $m\times n$ grid be the point configuration consisting of the lattice points in an $m\times n$ rectangular grid, and let a subdivision be a subdivision of this point configuration. For $n\geq 2$, denote by $D_{n-1}$ the $(n-1)$th central Delannoy number.

Central Delannoy formula. The number of subdivisions of a $2\times n$ grid is

$$2^{n-2}D_{n-1}.$$

The conjecture proposes a closed formula for the total number of subdivisions of a $2\times n$ grid, extending the small cases computed in the paper. Its status is not resolved in the supplied source.