It seems False-True has something rather interesting however:
w[0] and ws_offset are almost 100% correllated. Which is not very surprising as what we are observing 99% of the time is w[0]+ws_offset (with the other 1% being w[1]+ws_offset), so yes, there are two extra degrees of freedom there.
But - again, this to me looks like the simplest most standard multinomial glm you can make. So there must be a standard way of avoiding this happening, right?