Existence of cts-links distinguished by the parameterized invariant

A cts-link is a singular link considered in the paper, and $\widetilde{\Upsilon}_{x,y,z}$ and $\widetilde{\Upsilon}_{x,y,0}$ are the corresponding link invariants, with the latter obtained by setting $z=0$. The cts-link distinction conjecture. There exists a pair of cts-links that are distinguished by $\widetilde{\Upsilon}_{x,y,z}$ but remain indistinguishable under $\widetilde{\Upsilon}_{x,y,0}$. The conjecture would show that the parameter $z$ gives the invariant genuinely greater distinguishing power; no example establishing it had been found in the paper, so the claim remains open.