The Gram determinant conjecture for Möbius-band diagrams with one crosscap curve

Let $R=\mathbb{Z}[A^{\pm1},w,x,y,z]$. For $n\geq 1$, let $D_n^{(Mb)_1}$ denote the Gram determinant of type $(Mb)_1$, where the Gram matrix is formed from crossingless-connection diagrams on the Möbius band with the associated bilinear form. Let $d$, $w$, $x$, $y$, and $z$ be the parameters occurring in that form, and let $T_k$ and $S_k$ denote the Chebyshev polynomials used in the source. The Gram determinant conjecture. For $n\geq 1$,

$$D^{(Mb)_1}_n=\left[(d-z)((d+z)w-2xy)\right]^{\binom{2n}{n-1}}\prod_{k=2}^n\left(T_k(d)^2-z^2\right)^{\binom{2n}{n-k}}\prod_{k=2}^n(d^2-4)^{\binom{2n}{n-k}}\left(S_{k-1}(d)\right)^{2\binom{2n}{n-k}}.$$

The formula is presented as a conjectured closed form for the Gram determinant; the source also relates its final product factors to the determinant in the one-crosscap sector.