The packing and covering density conjecture for symmetric four-element sets

Let $a<b$ be coprime positive integers, and let

$$S=\{0,a,b,a+b\}.$$

For a finite set $S\subset\mathbb{Z}$, write $d_p(S)$ for its packing density and $d_c(S)$ for its covering density. The packing and covering density conjecture.

$$d_p(S)=\begin{cases}\frac{1}{4},&b-a\text{ is odd},\\[2pt]\frac{\lfloor(ab+b-a)/4\rfloor}{ab+b-a},&b-a\text{ is even},\end{cases} \qquad d_c(S)=\begin{cases}\frac{1}{4},&b-a\text{ is odd},\\[2pt]\frac{\lceil(ab+a+b)/4\rceil}{ab+a+b},&b-a\text{ is even}. \end{cases}$$

The statement gives the proposed exact packing and covering densities for symmetric four-element sets; the surrounding discussion identifies the related packing conjecture of Liu and Zhu as open and presents this covering counterpart as connected to a Ramsey-theoretic problem. The supplied source does not establish whether this combined statement has since been resolved.