Typically, its not about if a piece of hardware works with LPX.. It's whether or not it works with 10.8.x and if the Devs have written drivers that are compatible with 10.8.x that really matters....
In the case of Tascam, sadly, they don't have a good track record of providing support for newer OS X versions and instead just "end of life' their older products....
You will see from the following thread that the problem caused by Tascam not providing updated drivers for Mountain Lion, affects other DAWs Like PT and not just Logic...
http://duc.avid.com/showthread.php?t=342253
As for the 2400 and ML 10.8.4 itself...
At the moment, there doesn't appear to be anything new forthcoming from Tascam.... and they are not being overly helpful in regards to the non compatibility issues of the 2400 and ML 10.8.4 because as far as they are concerned they have "end of life"d that model...... Contact Tascam to register your wishes for a ML 10.8.4 driver.. Maybe if enough users do so, they might have a change of heart.. but based on their past history, I wouldn't hold my breath.
A few users have reported that they got their 2400 working under ML 10.8.2 but when asked how, none have been forthcoming with any answers.. The majorityof 2400 owners have stated its basically a 'no go' in regards to things like Fader movement etc... even using HUI compatibility mode... under Mountain Lion and have either moved on to new hardware that is fully supported under 10.8.x or have rolled back to 10.6.8 and Logic Pro 9 or whatever other DAW they use that works under Snow Leopard.
Finally, Apple made a mistake including the 2400 as a 'supported' hardware in the info for LPX and ML.... (They just copied the list over from the one they used with LP9 and Snow Leopard) They have since suggested that they will be updating the list and the 2400 will be removed from that list as Tascam are seemingly not intending to update their drivers for it any further....