I don't know of one resistor based passive that doesn't vary the output impedance based on the attenuation level. I think it is just the nature of the beast and as Andrew mentioned the LDR design is not immune to this (Side Note: The Truth Preamp does not vary the output impedance but that is because it uses active buffers on the outputs). It is one of the reasons why it is important to get the right impedance matching and in some cases to pay attention to cabling to attain the best sound from such a design. TVC passives have the same issue (see here, scroll down the page http://www.stevens-billington.co.uk/page102.htm) but are more forgiving. It was one reason I preferred TVCs earlier on before encountering the LSA.
Also, Ralph Karsten mentioned a number of times that the best sound from a passive will occur closest to the maximum point on the volume control (least attenuation). That is because the further around the dial you go the lower the output impedance. I'm never at much lower than 12 o'clock on my LSA and with my Atma-Sphere S-30 amps where I can vary the input sensitivity and gain, I'm never at less than 3 o'clock. So it's safe to say I'm operating at lower output impedance running into high input impedance amps (minimum 100k ohm, maximum 250k ohm).
Also, Ralph Karsten mentioned a number of times that the best sound from a passive will occur closest to the maximum point on the volume control (least attenuation). That is because the further around the dial you go the lower the output impedance. I'm never at much lower than 12 o'clock on my LSA and with my Atma-Sphere S-30 amps where I can vary the input sensitivity and gain, I'm never at less than 3 o'clock. So it's safe to say I'm operating at lower output impedance running into high input impedance amps (minimum 100k ohm, maximum 250k ohm).