Conjectured coefficients in the transfer-matrix expansion for Camassa–Holm peakons

Let $\tau_m(z)$ be the transfer matrices, let $\Lambda(z)$ be the spectral function, and let $\mathfrak h^{(k)}$ denote the Hamiltonians defined by the degree-$1$ terms of the transfer matrices. For each $m$, write

$$\tau_m(z) = \sum_{j=1}^m\Lambda(z)^j\sum_{0< k_j\leq k_{j-1}\leq\cdots\leq k_1,\,k_1+k_2+\cdots+k_j=m} a_j(k_1,k_2,\ldots,k_j)\\,\mathfrak h^{(k_1)}\mathfrak h^{(k_2)}\cdots\mathfrak h^{(k_j)},$$

with $a_1(m)=1$.

Conjectured coefficient formula. The coefficients satisfy

$$a_m(1,1,\ldots,1)=1,$$

and

$$a_{j+1}(m-j,\underbrace{1,\ldots,1}_{j})=\frac{m}{m-j},\qquad 0\leq j<m-1.$$

The other coefficients $a_j(k_1,k_2,\ldots,k_j)$ remain to be determined.

These formulas are suggested by the explicit transfer matrices through $m=7$; the remaining coefficients are not determined in the source, so the conjecture is open.