After ordering some CM109 based sound cards on the slow boat from China, the focus turned to minimizing the noise in whatever form factor it took, for future applications. Used the datasheet design for the AK4524 with an extra 4700uF, 5V regulator, RF choke, & power from an isolated bench supply. Also soldered a 10k chip resistor to the input, instead of the pot. It was a marvel of dead bug soldering.
That yielded 14 bits of precision. The 4700uF squeezed in a fraction of a 15th bit. Running it on the rated 5V instead of 3.3V definitely reduced the noise. There's some active power filtering inside the AK4524. Using too much or too little input resistance hurt it. Who knows how much of the noise is from the 10k as the analog source. Shorting it to ground produces all 0. There aren't many low impedance low noise sources.
The quest for the last 10 bits led to this nugget of information
written by the same Bonnie Baker you'll find in the famous TI video
Then, there's an equation which translates the SNR to the effective number of bits.
The AK4524 has 88dB on its datasheet for its most conservative SNR. ( 88 - 1.76 ) / 6.02 = 14
So basically, it only has 14 real bits of precision.