Fel's conjecture on universal symmetric polynomials for numerical semigroups

Let $S$ be the numerical semigroup under consideration, with multiplicity $m$, generators $d_1,\dots,d_m$, gap power sums $G_r(S)$, and invariants $K_p(S)$. Write $\sigma=(\sigma_1,\sigma_2,\dots)$ and $\delta=(\delta_1,\delta_2,\dots)$, where

$$\sigma_k=\sum_{i=1}^m d_i^k,\qquad \delta_k=\frac{\sigma_k-1}{2^k}.$$

Let $T_n$ be the universal symmetric polynomials defined from these power sums. Fel's conjecture. For every $p\geq 0$,

$$K_p(S)=\sum_{r=0}^p\binom{p}{r}T_{p-r}(\sigma)G_r(S)+\frac{2^{p+1}}{p+1}T_{p+1}(\delta).$$

This conjecture gives a complete formula for the invariants $K_p(S)$ in terms of the universal symmetric polynomials and the gap power sums; the displayed cases for $p\leq 3$ provide the initial evidence, while the general identity is the conjectural part.