Nonvanishing of Kirby stabilization factors ($p_{+}{p}_{-}={\mathcal{D}}^2$)

Let $H$ be the real Hopf pairing of Jones-Wenzl colors matrix, $d$ the dimension vector and $T$ the diagonal matrix of Jones-Wenzl twist eigenvalues. Twist balancing on the fusion of Jones-Wenzl colors gives $HTd=p_{+}{T}^{-1}d$, using ${\sum }_i{d}_i{N}_{ij}^k=d_j{d}_k$. Complex conjugation gives $H{T}^{-1}d=p_{-}Td$. Apply $H$ again and use finite sine orthogonality, $H^2={\mathcal{D}}^{2}I$. Since $Td\ne 0$, the displayed identity follows. Thus both factors used to normalize the Jones polynomial surgery invariant are nonzero.