Boyle–Ishai–Pass–Wootters toy conjecture on permuted Reed–Muller puzzles

Let $\mathbb{F}$ be a finite field with $|\mathbb{F}|=q\approx\lambda^2$. Let $p_1,\ldots,p_m$ be uniformly random polynomials of degree at most $\lambda$ in $\mathbb{F}[X]$, where $m=\lambda^{100}$, and let $q_1,\ldots,q_m$ be uniformly random functions from $\mathbb{F}$ to $\mathbb{F}$. Let $\pi\in S_{\mathbb{F}\times\mathbb{F}}$ be a uniformly random permutation. Define

$$S_i=\left\{\pi(x,p_i(x)):x\in\mathbb{F}\right\},\qquad T_i=\left\{\pi(x,q_i(x)):x\in\mathbb{F}\right\}.$$

Toy conjecture. The distributions $(S_1,\ldots,S_m)$ and $(T_1,\ldots,T_m)$ are computationally indistinguishable.

This simplified conjecture was proposed as a basis for candidate constructions of oblivious locally decodable codes. It is false: the paper gives an efficient algorithm distinguishing the two distributions, and independently Boyle, Holmgren, Ma, and Weiss obtained similar results.