The partite folklore clique decomposition conjecture

Let $(V_1,\ldots,V_q)$ be a balanced $q$-partition of a $q$-partite graph $G$, with $|V_i|=n$ for every $i$. Define

$$\hat{\delta}(G):=\min\{d(v,V_j):1\leq j\leq q,\ v\in V(G)\setminus V_j\}.$$

Call $G$ partite-$K_q$-divisible when, for every $1\leq j_1<j_2\leq q$ and every $v\in V(G)\setminus(V_{j_1}\cup V_{j_2})$, one has $d(v,V_{j_1})=d(v,V_{j_2})$.

Partite folklore conjecture. For each integer $q\geq 4$, the following holds for sufficiently large $n$: if $G$ is a balanced partite-$K_q$-divisible $q$-partite graph on $qn$ vertices with

$$\hat{\delta}(G)\geq \left(1-\frac{1}{q+1}\right)n,$$

then $G$ admits a $K_q$-decomposition.

This is the partite counterpart of the folklore high-minimum-degree conjecture for $K_q$-decompositions and is connected to the existence of mutually orthogonal Latin squares. The source presents it as open.