The Core has run multiple 10X libraries without problems. Please remember that these libraries are SINGLE-indexed, with an inline barcode at the beginning of Read 1 and indexes in the i7 position. Also, each sample has a SET OF FOUR i7 indexes (or sometimes more depending on kit version).
We will need to know all the indexes in order to demultiplex them correctly. The Core will have to add at least 5% of base-balancer (phiX) to avoid any problems caused by lack of sequence diversity in the polyA tails.