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

Let $Mb_{n,1}$ be the set of crossingless-connection diagrams between $2n$ marked boundary points of the Möbius band, with exactly one curve intersecting the crosscap. Let $G_n^{Mb_{n,1}}$ be its Gram matrix, and let $\tilde{G}_n^{Mb_{n,1}}$ be obtained by substituting $y=0$ and $w=1$. Let $T_k$ denote the Chebyshev polynomials of the first kind. The Möbius-band one-crosscap determinant conjecture. For $n\geq 2$,

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

The determinant is known to divide the Gram determinant of type $(Mb)_1$, but the displayed closed formula was presented as a conjecture.