As with the XBOX One, eventually maybe. It's not, unlike previous stuff, talking about jailbreaking an embedded computer system (past gen consoles fall into the embedded category of computers, in the sense of being application specific examples). It's instead talking about taking a PC like (in the same way a Chromebook/Chromebox is) and finding a way to make it boot up another OS (i.e. a live Linux distro), then you'd be in a position to start (from the inside) working on gaining access to the hypervisor controlled OS installation(s) - installations, since there's a main OS (different ones in each of the XB1 and PS4), and there's a 'game' os and both kinda run behind a hypervisor, so whether you are in the main os or the game os (or one is in focus as you actively use it) - you're really talking about actually using a VM containing the OS environment - so essentially two VM's controlled by a common hypervisor.
So you are talking contents of the OS and 'game' OS installations being in virtual HDD's stored on the actual HDD - and given these will be encrypted, it's not even simply a matter of extracting the physical HDD and cloning it and trying to decrypt the VHD's on binary image of the physical HDD (although, it's one approach that can deliver some clues). Chances are, the NG consoles have a TPM (Trusted Protection Module) similar to what's used on commercial grade PC's and Chrome devices, so chances are the VHD encryption is tied to the identity of the console's TPM (assuming there is one).
So all in all, and chances are i have wildly oversimplified things in the above thoughts, it's either going to be a very long road to a 'jailbreak', or if a serious flaw shows itself very rapidly during initial hacking attempts - then it could be a shortish road. But i suspect the 'shortish' road outcome is history, as the kind who really pull one out of the hat would have found that kind of flaw by now if it was there.
So irrespective of which NG platform you talk of, it's looking like a long haul followed by a longish haul, and then maybe some kind of building block found to create a shorter road to opening up the 'garden wall' in an exploitable form.
Now, technically, some may say the XB1 (being essentially based on Win10/Win Server OS base) will be the most vulnerable given Microsoft's history of Windows and it's vulnerabilities, but the Unix based OS in the PS3 could be no less secure if noone remembered to really lock it down - remember, Unix and Linux is only as secure as it's locked-down state. But i suspect with the amount of money tied up in the whole NG consoles projects, they won't have been sloppy with locking stuff down. With embedded stuff like the past-gen, half the security was in the relatively inaccessible nature of embedded device hacking and programming that kept them well out of the running for being hacked by kiddies and rank amateurs (in the embedded device world, hacking is the stuff of the very determined or those who work with developing for the embedded world).
So i can only say, hang in there if you are a believer - just don't literally hold your breath waiting on a short-cut to a solution, else you'll be six-foot under before they even get to step 1.1