The polynomial identity for the general rational Gaudin Lax matrix solution

Let $k\in\mathbb{N}$ and let $c_1,\dots,c_k$ be arbitrary constants. For each $i$, let $\{\vec{J}_i\}$ denote the set of vectors satisfying

$$\{\vec{J}_i\}=\{\vec{J}\in\mathbb{N}^i:J_1+2J_2+\cdots+iJ_i=i\}.$$

For $\vec{J}_i$, write $|\vec{J}_i|=\sum_{l=1}^iJ_l$ for its $1$-norm. The algebraic conjecture. The following polynomial identity holds:

$$\sum_{i=0}^k\left(\sum_{\{\vec{J}_i\}}\frac{c_1^{J_1}}{J_1!}\dots\frac{c_i^{J_i}}{J_i!}\lambda^{|\vec{J}_i|}\sum_{\{\vec{J}_{k-i}\}}\frac{c_1^{J_1}}{J_1!}\dots\frac{c_{k-i}^{J_{k-i}}}{J_{k-i}!}\mu^{|\vec{J}_{k-i}|}\right)=\sum_{\{\vec{J}_k\}}\frac{c_1^{J_1}}{J_1!}\dots\frac{c_k^{J_k}}{J_k!}(\lambda+\mu)^{|\vec{J}_k|}.$$

This identity is proposed as the algebraic step underlying the general analytic solution of the functional equations for the rational Lax matrix. The source supplies no resolution or further evidence for the conjecture, so its status remains open.