The optimal $d+1$ bound for fractional Helly in ultraproducts of $p$-adics

Let $K$ be an ultraproduct of the fields $\boldsymbol{Q}_{p_i}$ or $\boldsymbol{F}_{p_i}((t))$ over a non-principal ultrafilter, and let $\varphi(x,y)$ be a partitioned formula with $|x|\leq d$. The optimal fractional Helly bound conjecture. Every such formula satisfies $\operatorname{FHP}_{d+1}$; equivalently, the bound $2^d$ in the stated corollary can be improved to $d+1$. The paper proves the bound $\operatorname{FHP}_{2^d}$ and records $d+1$ as the expected optimal bound, so the improvement remains open.