Odd-order off-diagonally symmetric alternating sign matrix product formula conjecture

Let an off-diagonally symmetric alternating sign matrix (OSASM) be an alternating sign matrix invariant under reflection in the off-diagonal axis, and let $n$ be a nonnegative integer. Odd-order OSASM product formula conjecture. The number of $(2n+1)\times(2n+1)$ OSASMs is

$$|\mathrm{OSASM}(2n+1)|=\frac{2^{n-1}(3n+2)!}{(2n+1)!}\prod_{i=1}^{n}\frac{(6i-2)!}{(2n+2i+1)!}.$$

The formula is conjectured because the relevant Pfaffians have not been evaluated in this case; the paper notes that the equivalent Pfaffian identity remains difficult to prove.