Conjecture on the additive gap between integral and fractional $t$-clique covers

For a graph $G$, let $\mathcal{C}\mathcal{C}_t(G)$ and $\mathcal{C}\mathcal{C}^*_t(G)$ denote its integral and fractional $t$-clique cover numbers, respectively. Additive-gap conjecture. For every $\varepsilon>0$ and every positive integer $t$, there exists $n_0$ such that every graph $G$ on $n\geq n_0$ vertices satisfies

$$\mathcal{C}\mathcal{C}_t(G) \leq \mathcal{C}\mathcal{C}^*_t(G)+\varepsilon n^t.$$

The preceding argument establishes an analogous asymptotic statement for the decomposition number, but the supplied text gives no resolution of this clique-cover conjecture.