Okay, I've done a bit of research, primarily on Wiki (which we KNOW is always right. Right?). At a minimum, it's interesting background:
https://en.wikipedia.org/wiki/New_York_Tunnel_Extension" onclick="window.open(this.href);return false;
https://en.wikipedia.org/wiki/North_River_Tunnels" onclick="window.open(this.href);return false;
https://en.wikipedia.org/wiki/Marmaray" onclick="window.open(this.href);return false; (included as an example of recent rail tunnel construction).
What amazes me is the tunnel profile that's shown on the first link. The river channel, obviously, is deeper at certain points than others; not a uniform depth. So the gradient profile of the crossing is a bit of a roller coaster. I wonder in comparison how the depth and profile of the Bosphorus Strait compares to the Hudson. I could NOT find what the width of the Hudson (North) River is at the crossing point, and what length the tunnel sections are. The immersed tube of Marmary is slightly less than a mile at 4551'. The George Washington by comparison is 4760'.
What I'm getting at is if the tunnels are to be built beneath the sediment/silt at the bottom of the river, but if the river is of sufficient depth that an immersed tube, assembled in sections to account for the depth profile, sitting on the surface of the river, is sufficiently deep for shipping, why not just "drop" a tube on the river bottom similar to Marmary?
We know Amtrak has spent money on access on the NY side (the "box") and preliminary design. We know NJT started construction for the discontinued ARC tunnel on the NJ side, albeit to line up North of Penn in a deep cavern, much different than the Amtrak NY Penn expansion. How much more design work remains to be done for the project to be "shovel" ready? What can they start NOW while waiting for funding to materialize (Cuomo did much of the replacement TZB project similarly, getting it started and then adding to it as funding was "kind-of" identified)?
To me, this should be a "start digging now for what we know" type of project, because it could turn out much like ESA for the LIRR; delays, delays, delays. Again, Cuomo's approach to the TZB got it started quickly, and it will finish in 2017). If you at least get increments started, the tough stuff will be in preparation already.