The prime-level weight-three dimension conjecture

Let $p\geq 5$ be prime, and let $d(3,p)$ denote the dimension of the $\Q$-space of weight-three multiple polylogarithm values of level $p$. Standard relations are the regularized double shuffle, distribution, seeded, and lifted relations used in the paper.

Prime-level weight-three conjecture.

$$d(3,p)\leq \frac{p^3+4p^2+5p+14}{12}.$$

Moreover, equality holds if standard relations produce all the linear relations. The bound is inferred from computations for $p=5,7,11,13$; the source presents the assertion as a conjecture and gives no resolution.