The clique conjecture for unique-chromatic colourings

Given an edge-colouring of $K_n$, a copy of $H$ is unique-chromatic if some colour occupies exactly one edge of it. An edge-colouring is $H$-unique if every copy of $H$ is unique-chromatic, and $u_H(n)$ is the least number of colours in an $H$-unique edge-colouring of $K_n$.

Clique uniqueness conjecture. For every positive integer $t\geq 2$,

$$u_{K_t}(n)=n^{o(1)}.$$

The paper proves a polynomial lower bound for every non-complete graph without isolated vertices, so cliques are conjectured to be precisely the graphs admitting sub-polynomially many colours. The cases $t=4$ and $t=5$ are proved; the remaining clique cases are open.