Hello Allan! In my opinion you could try to re-read the article more carefully! Even at the first reading I had difficulty fully understanding the content. Now I'll explain the components well, then try to read it again:
The aim of the experiment is to understand whether the interaction with the nuclear lamina is related to gene silencing.
You have a two-component inducible system:
1) a vector containing Tk-hyg (promoter of the herpes virus timidine-kinases and gene for hygromycin resistance) and an array of LacO operators, this is the inducible component
2) a retroviral vector (which then integrates into the genome) that contains GFP (reporter) LacI (which when tying LacO allows the reporter to be viewed)
You have two variants of the second component:
1) one is used as "control", GFP-LacI
2) the other contains the C-terminal segment of the emerina (Δ is put in the name to make you understand that it is only a segment!), this segment allows the entire piece of genome that contains the vector to be recruited on the lamina.
You use these two variants because:
if it is true that the lamina-tiled genes are silenced LacI and LacO will not be able to interact and therefore you will not see the reporter, but you need to see that the inducible system works and then use the control (GFP-LacI) because this is not integrated near the lamina and so you are able to see the reporter!.