The unrestricted upper-bound conjecture for the total 2-coalition number

Let $G$ be a graph with minimum degree $\delta=\delta(G)$ and maximum degree $\Delta=\Delta(G)$, where $delta eq 2$. The unrestricted upper-bound conjecture states that

$$\operatorname{TC}_{2}(G)\leq \left\lfloor\frac{\delta}{2}\right\rfloor\left(\Delta-2\left\lfloor\frac{\delta}{2}\right\rfloor+1\right)+\left\lceil\frac{\delta}{2}\right\rceil.$$

The authors report that the restriction $\Delta(G)\geq 4\lfloor\delta(G)/2\rfloor-2$ from the preceding theorem may be omitted, and that no counterexample was found; the statement is known when $\delta(G)\leq 5$.