Sharpness of standard and non-standard relations for multiple polylogarithm values

Let $N$ be a level. Call $N$ standard if either $N=1,2,3$ or $N=p^n$ for a prime $p\ge 5$ and an integer $n\ge 1$; otherwise call it non-standard. Let $d(w,N)$ denote the dimension of the space of multiple polylogarithm values of weight $w$ and level $N$, and let $\operatorname{MPV}(w,N)$ denote that space. The standard relations are the relations already used to obtain the standard bounds, while non-standard relations are relations beyond them. Sharpness conjecture. If $N$ is a standard level, then the standard relations always provide the sharp bounds for $d(w,N)$, so that all linear relations can be derived from the standard ones. If $N$ is a non-standard level, then the bound in Corollary 5.25 of the cited work is sharp, and non-standard relations exist in $\operatorname{MPV}(w,N)$ for all $w\ge 3$, and in $\operatorname{MPV}(2,N)$ if $N\ge 10$. This conjecture predicts that the currently known standard and non-standard relations completely determine the dimensions in the standard cases and that the stated additional relations occur in every non-standard case; the paper reports numerical and low-level evidence but does not establish the general claim.