Strong counterexamples to Mubayi's supersaturation conjecture in every uniformity

The supersaturation problem asks, for a fixed $r$-graph $\mathcal{F}$, for the minimum number of copies of $\mathcal{F}$ in an $n$-vertex $r$-graph with $\text{ex}(n,\mathcal{F})+q$ edges. Mubayi conjectured a local form of supersaturation under a stability hypothesis: if $\mathcal{F}$ is non-$r$-partite and stable, meaning roughly that the extremal $\mathcal{F}$-free construction is unique and all near-extremal $\mathcal{F}$-free $r$-graphs are close to it, then this minimum should be at least $qc(n,\mathcal{F})$, where $c(n,\mathcal{F})$ is the minimum number of copies created by adding one edge to the extremal $\mathcal{F}$-free $r$-graph. We disprove this conjectured local lower bound in every uniformity. For every $r\ge 2$ and every $K>1$, we construct a stable $r$-graph $\mathcal{F}$ such that, for all sufficiently large $n$ and every $1\le q\le \delta n$, there is an $n$-vertex $r$-graph with $\text{ex}(n,\mathcal{F})+q$ edges and at most $K^{-1}qc(n,\mathcal{F})$ copies of $\mathcal{F}$. Thus the conjectured lower bound can already fail at $q=1$, and the failure can be by an arbitrarily large constant factor in every uniformity.