May tramping on the Routeburn - avalanches?

Hello, I apologize if this topic has been tackled before. I searched the threads and didn't see anything. I am hoping to tramp the Routeburn May 11 - 13th. I am a very experienced hiker and have hiked out-of-season in places like the Canadian Rockies and Bolivia. My concern is about avalanche risk. I realize that weather is variable and unpredictable at best, but can I expect trail closings or substantial risk at this time? Is it common to carry crampons and an ice-axe in May? I have contacted the DOC and they basically regurgitate the Routeburn winter tramping guidelines at me. They did say that once the first big snow dump occurs, it's avalanche season, they just didn't know if that would occur in May, June, or July. Anyone with experience tramping Routeburn in May? or attempted to tramp in May, but couldn't because of an avalanche closure? Also, I'm looking at alternatives if the Routeburn is closed. I was thinking of heading to the Kepler, but I'm not sure if it would be much better at this time of year. Other options include heading up to the North Island for the Tongariro Circuit. I have contacted transport companies and have found a company willing to transport to and from the Routeburn Shelter and the Divide on the dates mentioned. Although they also gave me a stout warning on hiking the Routeburn during this time of year. Any suggestions, info, etc would be greatly appreciated. Thanks.