Characteristic-$p$ dimensions of multihomogeneous components of $B_2$

Let $A_n(\mathbb F_p)$ be the free associative algebra on $n$ generators over the finite field $\mathbb F_p$, and let $B_2(A_n(\mathbb F_p))[m_1,\ldots,m_n]$ denote its multihomogeneous component of positive multidegree $(m_1,\ldots,m_n)$. Characteristic-$p$ dimension conjecture. If all $m_i$ are positive, then

$$\dim B_2(A_n(\mathbb F_p))[m_1,\ldots,m_n]= \begin{cases} 2^{n-2},&\text{if some }m_i\text{ is not divisible by }p,\\ 2^{n-1}-1,&\text{if every }m_i\text{ is divisible by }p. \end{cases}$$

This is presented as a consequence of the conjectural integral description of $B_2$ and predicts the precise characteristic-dependent dimension jump.