Tuple restrictions for zeta-like multizeta values

Let $(s_1,\dotsc,s_r)$ be a tuple of positive integers, and call it zeta-like when its associated multizeta value is zeta-like; call a tuple eulerian when its associated multizeta value is eulerian. Tuple restrictions. If $(s_1,\dotsc,s_r)$ is zeta-like, then

$$s_i\leq s_{i+1},\qquad (q-1)s_i\leq s_{i+1}\leq(q^2-1)s_i\quad (i=1,\dotsc,r-1),$$

and $(s_2,\dotsc,s_r)$ is eulerian while $(s_1,\dotsc,s_{r-1})$ is zeta-like. The second assertion can be iterated, implying that $s_i$ is even in the relevant characteristic-dependent sense for $i\geq2$.