A 2-plus-epsilon approximation for precedence-constrained single-machine scheduling with release dates

Consider the scheduling problem $1|\,r_j,\,prec\,|\sum w_jC_j$, consisting of precedence-constrained jobs on a single machine with release dates, where $C_j$ is the completion time of job $j$ and $w_j$ is its weight. A $(2+\varepsilon)$-approximation algorithm achieves a schedule whose total weighted completion time is at most $(2+\varepsilon)$ times optimal, for every $\varepsilon>0$.

Approximation conjecture. For any $\varepsilon>0$, there is a $(2+\varepsilon)$-approximation algorithm for

$$1|\,r_j,\,prec\,|\sum w_jC_j.$$

The paper's $\sqrt{e}/(\sqrt{e}-1)$-approximation is presented as unlikely to be the final guarantee; this conjecture proposes approaching a factor of $2$ while allowing an arbitrarily small additive $\varepsilon$.