Only one way to find out!

Wear it in the bath?

I've personally not used hair conditioner myself. I stick with Woolite and Downy Fabric Softener.

I wouldnt use anything that is not made for cleaning or improving the look of fabrics and faux fur on a fursuit :P
This includes all bodycare products, because they are made for real hair and skin. It's not about how fake and real fur/hair looks similar, but about their chemical compositions.