It depends what you mean by "that".
USGODAE Argo Data Viewer page(pre-loaded to search in this smallish area for profiles in the last 3 weeks)
click "go"
scroll down to the list of profiles
click "profile preview"
Note this is not the case in the whole SPG, just a small area ~100NM x 200NM. Other areas in the SPG have a shallow fresher slightly colder layer on top.
Mostly it only goes to ~1800m depth and then there's a salty layer, only two profiles from two different floats show constant density to 2000m+: R4901747_175 and R4902396_017.
Only the one profile shows constant spiciness to 2000m+ as well (suggesting actual deep mixing), but others show this to ~1800m.
I see the ex-gulf stream water you're talking about creeping up just off the continental shelf, and the sea surface height gradient leading directly to the area of these Argo profiles. But as I understand it (not an expert) the sea surface height is lower precisely because the water is denser there. The 0 dbar surface has a gradient, but below a certain depth the equal pressure surface has a gradient the opposite way. The place where most Atlantic water enters the SPG is over the mid atlantic ridge SW of Iceland where the ocean is less than 1000m deep so that deep gradient has no influence. (there's also the Coriolis force hence the gyre and energy needs to dissipate for that SSH bowl to fill up)
That said, that spot of ex-gulf stream water is right over the deepest part at 4000m, and I don't know much about what's going on below 2000m i.e. half of the water column at that spot. I've seen some rare deep Argo profiles in the SPG which show the water decreasing to ~2C and becoming slightly fresher towards the sea floor, but I don't know how it compares with atlantic deep water (I just assume there would be negligible difference) and I don't know how to filter for just deep profiles.