Both Apt-x and LDAC promise 'near CD-quality' BT transmission.
This is incorrect. Only AptX promises near Redbook CD quality. LDAC promises hi-res quality (up to 24 bit / 96 kHz).
In my experience both codecs deliver what they promise.
As far as the audible differences between the two, you won't discern it with just any material, especially with highly compressed masters.
However, having performed ABX testing (non-blind) between AptX with MDR-1RBT MK2 vs MDR-1ABT using the same high dynamic range DSD material, LDAC is clearly ahead in terms of definition and clarity of individual instruments, as well as other subtleties such as reverb decays.
You could argue that this can potentially be attributed to the differences between the two headphones, but being very familiar with the sound of each one (I still own both) I already took this into account. Besides, while the 1ABT is undoubtedly the better sounding headphone of the two (using the same codec), anyone who has heard both will tell you that they sound very similar. In fact, were it not for the LDAC support, I would not have been able to justify upgrading my 1RBT MK2.
TBH I don't really believe Sony's hype about LDAC. Apt-x sounds great anyway, and is a clear improvement on SBC.
LDAC is not a religion, so does not require belief
Thankfully, it's fairly easy to test it if you can get a hold of a compatible source (e.g. a Sony Walkman or a recent Xperia device).
I agree that AptX already sounds great and it made bluetooth music a viable option for many people (it's a shame that many car manufacturers still don't support it though). However, LDAC is also a "clear improvement" on AptX. Although, it may be past the point of diminishing returns for some people, especially those who don't listen to anything higher than 320 kbps MP3 or equivalent, and / or over-compressed modern recordings.