Power conditioner wire gauge? Serious issue

Your power conditioner should say how much power it can supply. As long as you do not exceed that it is not a problem.

If you run 6 gauge to a sub panel for my stereo should you use 6 gauge for the power conditioner???

When supplying power to a conditioner, the wall needs to have the capacity to do all the units fed by the conditioner but each outlet out of the conditioner only needs to have a capacity for the one unit it serves.  The travel loss can be collective as well as singular, which ever applies.  

Example, each lane in a freeway is sized for just one lane.  The on and off ramps are usually not the lane count of the freeway.   

The collective vs the singular circuits is the need.  There are units powered by a conditioner like a DAC that may only draw one amp.  or less.  Then there are loads fed by the conditioner that draw 10 amps. Collectively is just that, but all circuits individually do not draw the collective load. 

My Audience power conditioner has a 10 AWG power cord, but everything inside is 12 and 14 gauge.

If you think 16awg is small…take a look at the wire inside the fuse that is on the power conditioner…