Chirality of torus-covering $T^2$-links of degree three

A torus-covering $T^2$-link of degree $n$ is a surface-link consisting of tori, in the form of an unbranched covering of degree $n$ over the standard torus. We focus on a torus-covering $T^2$-link of degree 3, which is determined by a pair $(a,b)$ of 3-braids satisfying $ab=ba$, denoted by ${\mathcal{S}}_3(a,b)$. We investigate to what extent the chirality of ${\mathcal{S}}_3(a,b)$ is detected by invariants such as the triple linking numbers, the number of Fox $p$-colorings, and the quandle cocycle invariant associated with $p$-colorings. In particular, we determine the quandle cocycle invariant for ${\mathcal{S}}_3(a,b)$ associated with tri-colorings.