Low-dimensional sum-set decomposition conjecture for folded Reed–Solomon list recovery

Let $d4aa$ be a folded Reed–Solomon code of rate $R$, with folding parameter $s$, and let $d4a1_1,\ldots,d4a1_n$ be subsets of the alphabet of size $d4ce$. For a positive integer $\ell$, let an $(r,\ell)$ sum-set mean a sum-set with the parameters indicated in the source. For every $\ell\in\mathbb N$, $b\varepsilon>0$, and $R>0$, there is an $s_0=\Omega(\ell/\varepsilon^2)$ such that, for every rate-$R$ $s$-folded Reed–Solomon code $d4aa$ with $s>s_0$, the list

$$\{c\in\mathcal C\mid \Delta(c,L_1\times L_2\times\cdots\times L_n)<1-R-\varepsilon\}$$

is exactly the union of $\poly(\ell/\varepsilon)$ many $(O((R+\varepsilon)/\varepsilon),\ell)$ sum-sets $P_1,\ldots,P_t$:

$$\{c\in\mathcal C\mid \Delta(c,L_1\times L_2\times\cdots\times L_n)<1-R-\varepsilon\}=\bigcup_{i=1}^tP_i.$$

Low-dimensional sum-set decomposition conjecture. For every choice of $\ell\in\mathbb N$, $\varepsilon>0$, and $R>0$, the decomposition above exists for all sufficiently large folding parameters $s$, with $s_0=\Omega(\ell/\varepsilon^2)$ and only $\poly(\ell/\varepsilon)$ sum-sets. This would give a combinatorial list-recovery bound of $\poly(\ell/\varepsilon)\cdot\ell^{O(R/\varepsilon)}$, stated in the source to be optimal. The assertion is strong because it requires exact equality with a union of a few low-dimensional sum-sets, rather than mere containment.