The plus-symmetric alternating sign matrix factorization conjecture

Let $P_n(x)$ be the generating function for alternating sign matrices invariant under flips in the vertical and horizontal axes, with the weight defined in the source. Such matrices have odd size. Let $T_n(x,0)$ and $T_n(x,1)$ be the triangular-array generating functions defined in the source. Plus-symmetric factorization conjecture. For $n\geq1$, it is conjectured that

$$P_{4n+1}(x)=T_n(x,1)T_n(x,0),$$

and

$$P_{4n-1}(x)=T_{n-1}(x,1)T_n(x,0).$$

The source states that these formulas have been verified only for small $n$, so their general validity remains open.