The cyclotomic anti-involution fixes the affine and Murphy generators

On the free associative algebra define $\ast$ by $r^{\ast }=r$ for $r\in R$, $T_i^{\ast }=T_i$, $X_j^{\ast }=X_j$, $(X_j^{-1}{)}^{\ast }=X_j^{-1}$, and $(ab{)}^{\ast }=b^{\ast }{a}^{\ast }$, extending $R$-linearly. Word reversal is an involutive anti-automorphism, so this defines an involutive anti-automorphism of the free algebra.

The quadratic relation is a polynomial in one fixed generator and is fixed by reversal. Each braid word in the $T_i$ is palindromic, each distant commutation relation reverses to the same equality with sides exchanged, each relation $X_j{X}_k=X_k{X}_j$ and $T_i{X}_j=X_j{T}_i$ reverses to its equivalent reversed equality, and $T_i{X}_i{T}_i=Q{X}_{i+1}$ is fixed because its left word is palindromic and $Q$ is central. The inverse relations reverse to the other inverse equations, and the cyclotomic relation $p(X_1)=0$ is fixed because it is a polynomial in $X_1$. Thus reversal sends every defining relator into the defining ideal. As the ideal is two-sided, it is preserved under reversal, so $\ast$ descends to an involutive anti-automorphism of $H_{r,n}$.

The quotient is generated by the displayed letters and $R$, so any anti-homomorphism fixing them is unique. By [F3], $L_j=X_j$, hence $L_j^{\ast }=L_j$. On words in the finite generators $T_i$, reversal sends a reduced word for $w$ to the reversed reduced word for $w^{-1}$; therefore $T_w^{\ast }=T_{w^{-1}}$, as for the finite reversal anti-involution.