Antidiagonal-permutation dominant-coefficient conjecture

For $n\geq 1$, define the reversal permutation

$$\pi_n \coloneqq \prod_{i=1}^{\left\lfloor n/2\right\rfloor}(i,n-i+1)\in\mathfrak{S}_n.$$

Let $r_n^{\pi_n}$ be the corresponding element of $\mathfrak{U}_{\mathsf{DY}}^1$, and let $p_{n,m}$ denote the coefficient of $r_{n+m}^{\pi_{n+m}}$ in the product $r_n^{\mathsf{id}}\circ r_m^{\pi_m}$. Antidiagonal-permutation dominant-coefficient conjecture. The identity

$$r_1^{\mathsf{id}}\circ r_n^{\pi_n}=-n\,r_{n+1}^{\pi_{n+1}}+r_n^{\pi_n}\star r_1^{\mathsf{id}}+\sum_{i=1}^n r_{n+1}^{\pi_{n+1}\circ(i,i+1)}$$

holds in $\mathfrak{U}_{\mathsf{DY}}^1$; moreover, $p_{n,m}$ is the dominant coefficient of $r_1^{\mathsf{id}}\circ r_n^{\pi_n}$, and

$$p_{n,m}=(-1)^n\bigl(\lvert p_{n-1,m}\rvert+\lvert p_{n,m-1}\rvert\bigr)$$

with $p_{1,1}=-1$, $p_{1,m}=-m$, and $p_{n,1}=0$ for $n>1$. This conjecture proposes a distinguished family of leading coefficients and a recurrence for them; the supplied text gives no evidence of resolution.