The bound on the number of sign vectors associated with an input

Let $n$ be a positive integer, let $P_x$ be the region associated with a sign vector $x\in\{-1,+1\}^n$, and let $(C,c)$ be a quadratic-programming input. The preceding proposition shows that an input may belong to multiple regions $P_x$. The region-count conjecture. For every input $(C,c)$, the membership relation $(C,c)\in P_x$ holds for at most $n$ distinct sign vectors $x\in\{-1,+1\}^n$. This observation is based on computations reported in the source, which found at most $n$ such sign vectors; the supplied text does not establish the bound theoretically.