This problem was proved to be NP-complete in 

* J. Nešetřil, V. Rödl, _Complexity of diagrams_, Order **3** (1987) 321–330, https://doi.org/10.1007/BF00340774, 

but a mistake was discovered, and later corrected, for a simple proof and brief history, see 

* G. Brightwell, _On the complexity of diagram testing_, Order **10** (1993) 297–303, https://doi.org/10.1007/BF01108825.


I've asked the same question myself earlier this year from some people working in the field, the above links were given to me by Piotr Micek. Some papers to read are also given in 

* Janos Pach, Istvan Tomon, _Coloring Hasse diagrams and disjointness graphs of curves_, https://arxiv.org/abs/1908.08250v1 

on page 2, before Theorem 1. Also see 

* B. Bollobás, G. Brightwell & J. Nešetřil, _Random graphs and covering graphs of posets_, Order **3** (1986) 245–255, https://doi.org/10.1007/BF00400288.