The determinant conjecture for the remaining Möbius-band block

Let $n\geq 2$, let $\widetilde{G}_{n}^{Mb_{n,1}}$ denote the remaining block in the decomposition of the Gram matrix of type $(Mb)_1$, and let $T_{2k}$ be the Chebyshev polynomial of the first kind. Remaining-block determinant conjecture.

$$\det\bigl(\widetilde{G}_{n}^{Mb_{n,1}}\bigr)=\prod_{k=2}^n\bigl(T_{2k}(d)-2\bigr)^{\binom{2n}{n-k}}.$$

The paper states that this product is conjectured to be a factor of $D_n^{\mathit{Mb}}$ and that the assertion implies the preceding $(Mb)_1$ Gram determinant conjecture via the proved theorem. Its general validity remains open.