Strong counterexamples to Mubayi's supersaturation conjecture in every uniformity

The classical supersaturation problem, originating in classical results of Rademacher and Erdős on complete graphs, asks for the minimum number of copies of an $r$-graph $\mathcal{F}$ in an $n$-vertex $r$-graph with $\ex(n,\mathcal{F})+q$ edges. Mubayi conjectured that, for every stable non-$r$-partite $r$-graph $\mathcal{F}$, this minimum is at least $q c(n,\mathcal{F})$, where $c(n,\mathcal{F})$ is the minimum number of copies created by adding one edge to the $n$-vertex extremal $\mathcal{F}$-free $r$-graph. Ma and Yuan recently constructed infinitely many graph counterexamples, with arbitrary chromatic number at least four.
We construct, for every $r\ge 2$ and every $K>1$, a stable non-$r$-partite $r$-graph $\mathcal{F}$ with the following property: for all sufficiently large $n$ and every integer $q$ with $1\le q\le \delta n$, there is an $n$-vertex $r$-graph with $\ex(n,\mathcal{F})+q$ edges and at most $K^{-1}q c(n,\mathcal{F})$ copies of $\mathcal{F}$. Thus Mubayi's conjectured lower bound can already fail at $q=1$, and the failure can be by an arbitrarily large constant factor in every uniformi