Weight restrictions for eulerian and zeta-like multizeta values

Let $p$ be the characteristic, let $q$ be the size of the finite field, and let a tuple have depth $r$. Its weight is the sum of its components; call a tuple primitive when it is not obtained from another tuple by the relevant common power operation. Weight restrictions. The source conjectures: (1) an eulerian multizeta value in depth $r>1$ can occur only in weights $p^m(q^k-1)$, with primitive ones only in weights $q(q-1)$ or $q^n-1$ for $q>2$, and in weights $2^n-1$ and $2^n$ when $q=2$; (2) when $q=p$ and $r>1$, the weight of a zeta-like but non-eulerian tuple is $p^m$ times a number with no zero digit and at most one digit equal to $1$; (3) in depth $r$, the smallest weight of a zeta-like value is $q^{r-1}$; (4) for $q>2$, the smallest weight of an eulerian value is $q^r-1$, $(q-1)q$, or $q-1$ according as $r>2$, $r=2$, or $r=1$; and (5) weight $q^k$ is not the weight of a primitive zeta-like tuple if $k>r>3$, nor if $k>3$ when $r=3$ or $r=2$.