Ishii–Morton Gram determinant conjecture for the Möbius-band type $(Mb)_1$

Let $n\geq 1$, let $D_n^{(Mb)_1}$ denote the Gram determinant of type $(Mb)_1$, let $T_k$ be the $k$th Chebyshev polynomial of the first kind, and set $d=-A^2-A^{-2}$. Ishii–Morton's conjecture. The determinant is

$$D_n^{(Mb)_1}=\left[(d-z)((d+z)w-2xy)\right]^{\binom{2n}{n-1}} \prod_{k=2}^n\bigl(T_k(d)^2-z^2\bigr)^{\binom{2n}{n-k}} \prod_{k=2}^n\bigl(T_{2k}(d)-2\bigr)^{\binom{2n}{n-k}}.$$

The paper attributes this conjecture to the cited source and proves a theorem reducing it to the determinant of a remaining block, but the closed formula itself is not established in general.