Erdős–Sauer conjecture: decompose every $r$-uniform hypergraph into few cliques and single edges (Erdős #719)

Fix $r\geq 2$. Write $K_r^r$ for a single $r$-edge (the complete $r$-uniform hypergraph on $r$ vertices) and $K_{r+1}^r$ for the complete $r$-uniform hypergraph on $r+1$ vertices, which has $r+1$ edges. Let $\mathrm{ex}_r(n;K_{r+1}^r)$ denote the maximum number of $r$-edges that can be placed on $n$ vertices without forming a $K_{r+1}^r$. Question: is every $r$-uniform hypergraph $G$ on $n$ vertices the union of at most $\mathrm{ex}_r(n;K_{r+1}^r)$ copies of $K_r^r$ and $K_{r+1}^r$, no two of which share a $K_r^r$ (i.e. no two of the chosen pieces have a common $r$-edge)? Equivalently, can the edge set of every $r$-uniform hypergraph on $n$ vertices be partitioned into at most $\mathrm{ex}_r(n;K_{r+1}^r)$ pieces, each of which is a single $r$-edge or the $r+1$ edges of an $(r+1)$-vertex clique?

Acceptance. FULLY RESOLVES: a complete proof of the Erdős–Sauer conjecture — that every $r$-uniform hypergraph on $n$ vertices edge-decomposes into at most $\mathrm{ex}_r(n;K_{r+1}^r)$ pieces, each a single edge or an $(r+1)$-vertex clique with no two pieces sharing an edge — for all $r\geq 2$ and $n$ (machine-checkable preferred, otherwise a full written proof); OR an explicit counterexample: specific $r$, $n$, and hypergraph $G$ that provably requires strictly more than $\mathrm{ex}_r(n;K_{r+1}^r)$ such pieces, supplied with a verified value of the Turán number $\mathrm{ex}_r(n;K_{r+1}^r)$ and a proof of the lower bound on the number of pieces. ADVANCES (each independently checkable): (a) prove the conjecture for a fixed small $r$ (e.g. $r=3$) or an infinite family of $r$; or (b) verify it exhaustively for all $r$-uniform hypergraphs on $n\leq N$ vertices for a new record $N$, with the search code, the verified $\mathrm{ex}_r$ values used, and an exhaustiveness certificate. Deliver the proof, the counterexample with certificate, or the verification code plus the attained range.