Kim's Italian domination conjecture for Cartesian products of directed cycles

Let $C_n$ denote the directed cycle of order $n$, and let $b3_I(D)$ be the Italian domination number of a digraph $D$. For an odd integer $n$, Kim's conjecture.

$$\gamma_I(C_4 \Box C_n) = 2n + 2.$$

This conjecture concerns Italian domination in the Cartesian product of a directed $4$-cycle and an odd directed cycle. The source states the conjecture but provides no resolution, so its status remains open.