The diameter-three $\Omega_\omega$ upper-bound conjecture

Let $G$ be a graph of diameter $3$ with $n$ vertices, and let $\Omega_\omega(G)$ denote the $\omega$-cover pebbling parameter used in the paper. For $\omega\geq 1$, diameter-three $\Omega_\omega$ upper-bound conjecture.

$$\Omega_\omega(G)\leq\left\lfloor\frac{3}{2}(n-2-\omega)+1\right\rfloor.$$

The authors present this as an analogue of their diameter bounds for $\psi$ and state that good diameter bounds for $\Omega_\omega$ appear harder to establish; the conjecture remains open in the supplied text.