Square-root error conjecture for the counting norm-trace problem in finite semisimple algebras

Let $B$ be a semisimple algebra of dimension $n$ over $\q$. For $a\in \q^*$, let $x\in B^*$ be regular and let $r\geq 2$ be an integer. Writing

$$B=M_{d_1}(\mathbb{F}_{q^{n_1}})\times\cdots\times M_{d_k}(\mathbb{F}_{q^{n_k}}),$$

so that $n=d_1^2n_1+\cdots+d_k^2n_k$, define $N(B,r,x,a)$ to be the number of $(g_1,\ldots,g_r)\in(B^*)^r$ such that $g_1\cdots g_r=x$ and ${\rm Tr}_B(g_1+\cdots+g_r)=a$. Square-root error conjecture. One has

$$\left|N(B,r,x,a)-\frac{|B^*|^{r-1}}{q}\right|\leq r^{d_1+\dots+d_k}q^{\frac{(r-1)n-1}{2}}.$$

The estimate is the counting analogue of the norm-trace estimate proved earlier in the paper. The expected main term is $|B^*|^{r-1}/q$, and the asserted bound represents a square-root error for regular $x$; the source presents this as a hoped-for estimate rather than a proved result.