But Elizabeth asks V to lie and to NOT tell her husband. Not the other way around :-P you should correct that

Made a post in a wrong article like an idiot.
But found Mr. Blue Eyes, spying on you and Jeff in the Park. You can't shoot or interact with Mr Blue eyes, or for that matter reach him in any way it seems.
If you look into the computers you find along the way, it mentions BLUE, ORANGE and such.

You can acually repair the TV monitor on the other side of the monitoring room if you have 16 Technical Ability, but I can't recommend it cause V and Johnny gets a dose of the Peralezes medicine, if you catch my drift.

You can also try to call Jefferson directly after hacking the van, but he simply redirects you to Elizabeth.
Also (I'm assuming this is triggered by fixing the screen in the apartment and getting a dose of neural reprogramming) you will get an anonymous call before the final meeting telling you to mind your own business.
Connecting this to Sandra Dorsett questline and Operation Carpe Noctem datashard, it seems obvious the Night Corp is the baddie here.

anyone knows the name of the music playing during the conversation with Elizabeth ?

Clair de lune

The concealed apartment part is a non-essential option, doesn’t progress the dialogue, the essential one is “They’re conditioning you, remotely”, if you wanna tell Jefferson the truth.

Interesting part of this mission I just discovered. If you get to the Maelstrom hideout in time you can you can actually see two veteran guards exit the van then climb up the building towards the roof on the right where a cloaked transport craft is waiting to pick them up and evac them.
One of these guards carries an encrypted note that is suspending all operation of "Project Condin" with team black being the last to withdraw. Unfortunately it doesn't give any more information as to who is behind this.

There is a similar 'unknown device' on top of the mental hospital from Cuckoo's Nest gig.