Empty-intersection conjecture for copies of complete graphs above the Turán threshold

Let $V_1,V_2,\ldots,V_{k-1}$ be the vertex classes of the balanced Turán graph $T_{k-1}(n)$, with sizes satisfying $\left\lceil\frac n2\right\rceil\geq |V_1|\geq\cdots\geq |V_{k-1}|\geq\left\lfloor\frac n2\right\rfloor$ and $\sum_i|V_i|=n$. Let $T_{k-1}^{\sqsubset}$ be obtained from $T_{k-1}(n)$ by adding an edge $\{x,y\}$ in $V_1$, adding an edge $\{u,v\}$ in $V_2$, and deleting the edge $\{u,x\}$. Empty-intersection conjecture. If a graph on $n$ vertices has $t_{k-1}(n)+1$ edges and the copies of $K_k$ have empty intersection, then the number of copies of $K_k$ is at least the number in $T_{k-1}^{\sqsubset}$, namely

$$(|V_2|-1)|V_3|\cdot|V_4|\cdots|V_{k-1}|+(|V_1|-1)|V_3|\cdot|V_4|\cdots|V_{k-1}|=(|V_1|+|V_2|-2)|V_3|\cdot|V_4|\cdots|V_{k-1}|.$$

This would generalize the paper's theorem for triangles, but the source gives no resolution and presents it as an open problem.