The clique counterexample conjecture for balanced edge-colorings

Let $S(N)$ be the set of natural numbers $q$ such that $4\leq q\leq N$ and, for every $n_0$, there is some $n\geq n_0$ with $n\equiv 1\pmod{\ell}$ and a balanced coloring of $K_n$ in $\binom{q}{2}$ colors containing no rainbow copy of $K_q$. Clique counterexample conjecture. For every natural number $N$,

$$S(N)=\{n\in\mathbb{N}:n\geq 4\}.$$

This conjecture asserts that the Erdős–Tuza question has a negative answer for every clique of size at least four. The paper proves that $|S(N)|=N-(1+o(1))N/\log N$, showing that most clique sizes have the required counterexamples, while the full assertion remains open.