Affine subspace hereditary properties are testable

Let $p$ be a prime, and let an affine-invariant property $\mathcal{P}$ assign to functions $f:\mathbb{F}_p^n\to [R]$ a property that is preserved under affine transformations. The property is affine subspace hereditary if, whenever $f$ satisfies $\mathcal{P}$, its restriction to every affine subspace of $\mathbb{F}_p^n$ also satisfies $\mathcal{P}$. Affine subspace hereditariness conjecture. Every affine subspace hereditary property is testable with $1$-sided error. This conjecture proposes that affine subspace hereditariness is the minimal structural condition needed for one-sided testability of affine-invariant properties; the paper presents progress toward it and reduces the question to testability of properties defined by induced affine constraints.