Constant-depth Frege oddtown conjecture for nontrivial prime moduli

Let $F_{d}$ be the depth-$d$ fragment of Frege, let $oddtown_{k}$ be the oddtown axiom scheme, and let $Count^{p}_{n}$ be the modular counting principle modulo $p$. For a proof system $P$, write $P\vdash_{poly(n)}\varphi_{n}$ when each formula $\varphi_{n}$ has a polynomial-size $P$-proof.

Constant-depth Frege oddtown conjecture. For each $d\in\mathbb{N}$ and each prime $p\neq 2$,

$$F_{d}+oddtown_{k}\not\vdash_{poly(n)} Count^{p}_{n}.$$

The preceding result establishes the converse direction for powers of $2$ at the level of $V^{0}$. This conjecture strengthens the proposed separation to constant-depth Frege proof size; the paper does not prove it in full.