Sustainable fashion will undoubtedly continue to gain traction in coming years. H&M's Conscious Collection plays a major role in raising awareness of sustainable fashion, yet their overall business model will preclude that organization from becoming truly sustainable.
Contrary to some previous comments, many sustainable fibers have a better look and texture than the synthetics and blends that have come to mark much of the fashion industry in recent decades; this is especially true when comparing to H&M's (and other fast fashion retailer's) standard lines in which the fabric pills and breaks apart after a few washes. Fabrics including organic cotton, wool, silk, hemp-blends and others have greater longevity, and Tencel/Modal offers draping and texture found in synthetic rayon/viscose products without the environmental footprint.
Sustainable fashion designers are already addressing fabric innovations and style, and in my experience as a sustainable fashion writer and researcher, price will remain a primary barrier because consumers are not used to paying a complete or true cost of fashion. In addition, the rapid change of styles encouraged by many retailers will have to adjust so consumers do not feel the need to buy new pieces as frequently, or dispose of the old.
You cannot solve "sustainable fashion" with a "fast fashion" business model -- it's not only fabrics, style and texture, but quality, price and tempering the desire to consume.
However, in agreement with other comments, the Millennials are all over this, and are demanding sustainable, stylish, quality and lasting clothing. It's not an unsolvable problem, but it likely won't be solved by the likes of H&M or their Conscious Collection.