The random-graph clone conjecture for total difference labeling

Let $G$ be a random graph in the Erdős–Rényi model, and let $H$ be its clone. Write $\chi_{td}(G)$ for the total difference labeling number of $G$.

Random-graph clone conjecture. For every $\epsilon>0$, with probability $1$,

$$\chi_{td}(H) \leq (1+\epsilon)\chi_{td}(G).$$

This conjecture asks whether the general upper bound for the total difference labeling number of a clone is asymptotically close to equality for Erdős–Rényi random graphs. The supplied text gives no resolution evidence, so its status remains open.