Benjamin,
Two more suggestions:
Using the bq2425x charger family with bq51013B, we suggest using the HPA764 Rev C unseated, it is the most updated.
Also, Bq51013 is old part; use the bq51013B (wpc1.1 compliant) or at least a bq51013A if wpc1.1 is not needed.
The other solution is using the direct charging solution using the bq51050; the regulation voltage is 4.2 or 4.35V (bq51051B).
Thanks
Tahar