The conjecture on the complementary bounding function of the almost identity function

Let $g$ be the almost identity function

$$g(x)=\begin{cases} 3 & \text{for } x=2,\\ x & \text{for } x>2. \end{cases}$$

For a graph $G$, let $\alpha(G)$ be its independence number and let $\theta(G)$ be its minimum clique-cover number. The complementary bounding function $g^*$ is the optimal function bounding $\theta(G)$ in terms of $\alpha(G)$ for graphs whose chromatic number is bounded by $g(\omega(G))$. The conjecture on $g^*$.

$$g^*(x)=\left\lfloor\frac{8}{5}x\right\rfloor.$$

The statement is suggested by the preceding upper bound and is presented as a conjectural formula; the supplied text gives no resolution, so it is recorded as open.