The general Turán number conjecture for disjoint cliques in multipartite graphs

Let $r>t\ge 3$ and $k\ge 2$, and let $n_1,\dots,n_r$ be sufficiently large. For $I\subseteq [r]$, write $m_I:=\min_{i\in I}n_i$. For a partition $\mathcal P$ of $[r]$, write $n_I:=\sum_{i\in I}n_i$ and

$$n_{\mathcal P}:=\max_{I\in\mathcal P}\{n_I-m_I\}.$$

The general multipartite Turán conjecture. The Turán number of the disjoint union $kK_t$ in the complete $r$-partite graph is

$$\operatorname{ex}(K_{n_1,\dots,n_r},kK_t)=\max_{\mathcal P}\left\{(k-1)n_{\mathcal P}+\sum_{I\ne I'\in\mathcal P}n_In_{I'}\right\},$$

where the maximum is over all partitions $\mathcal P$ of $[r]$ into $t-1$ parts. This conjecture seeks a general formula extending the known results for earlier cases and the paper's theorem for $r=4$ and $t=3$; its validity for arbitrary $r>t\ge3$ remains open.