Hmm, now that's odd. I tried walking a base around 
and monitoring the sickel to see what happened to it. I walked it from 
up to 
, left to 
, down to 
and finally right to 
, crossing the STR boundary twice. I noticed nothing weird at the second boundary crossing...
EDIT: I've continued it further, to
and then back up to , and noticed nothing "singularitylike". However, I did find that on returning to , I was once again at a sickel between the two principal fixed points, only now they had been swapped. Wtf? Does this mean there's a merged tetrational using the two fixed points in opposite order? That'd be very bizarre. Or, perhaps, this in fact indicates that there is a failure somewhere (probably at the STR boundary) and the continuation is in fact not possible.
EDIT: I've continued it further, to