I don't think that there's even API for autoscaler to increase storage size of persisted disks in stateful sets, even if it's not advisable, even for something like mongodb. I guess that it would involve restarting the pod, too.
In GKE you can scale up your storage limit in your PersistentVolumeClaims underlying your existing PeristsentVolume and it will dynamically resize your existing volume. After (manually) POD restart, it should by bound to the same pvc and data stored in resized PV.
Hope this help.
Please share with your experience.