$C_4$-saturation conjecture for the complete four-partite graph

Let $K_4^n$ be the complete $4$-partite graph with $n$ vertices in each part, and let $C_4$ be the cycle of length four. Complete four-partite $C_4$-saturation conjecture. For every $n\geq 1$,

$$sat(K_4^n,C_4)=5n-1.$$

The paper previously gives the bounds $\lfloor 9n/2-1\rfloor\leq sat(K_4^n,C_4)\leq 5n-1$; the conjecture asserts that the upper bound is exact. No resolution evidence is supplied.