Arm-property conjecture for Kadar–Yu Gram determinants

Let $l\geq -1$, let $p\geq l+2$, let $m\geq 1$, and let $\lambda\vdash l+2$. Let ${\mathcal V}^{n}_{(p,\lambda)}$ and ${\mathcal C}^{n}_{(p,\lambda)}$ be the marginal vertex function and its Chebyshev expression for the arm of the Kadar–Yu graph, with $n=p+2m$.

Arm-property conjecture. The identity

$${\mathcal V}^{n=p+2m}_{(p,\lambda)}={\mathcal C}^{n=p+2m}_{(p,\lambda)}$$

holds for all $l\geq -1$, $p\geq l+2$, and $m\geq 1$.

This extends the arm property beyond the ranges established by direct calculation and would give a uniform recursive description of the relevant Kadar–Yu Gram determinants.