Wan's product-trace conjecture for finite semisimple algebras

Let $p$ be a prime and let $\mathbb F_q$ be the finite field of $q$ elements with characteristic $p$. A finite semisimple algebra over $\mathbb F_q$ has the form

$$B=M_{d_1}(\mathbb F_{q^{n_1}})\times\cdots\times M_{d_s}(\mathbb F_{q^{n_s}}),$$

with dimension $N=n_1d_1^2+\cdots+n_sd_s^2$. Write $B^{\times}$ for the group of units of $B$, and let $\operatorname{Tr}_B$ and $\operatorname{Nr}_B$ denote its reduced trace and reduced norm. For $x\in B^{\times}$, $a\in\mathbb F_q$, and $r\ge2$, define

$$\operatorname{N}(B,r,x,a)=\#\{(g_1,\ldots,g_r)\in(B^{\times})^r\mid g_1\cdots g_r=x,\ \operatorname{Tr}_B(g_1+\cdots+g_r)=a\}.$$

Here $x$ is called regular when it is a regular element of $B^{\times}$.

Wan's conjecture. For $a\in\mathbb F_q^{\times}$, regular $x\in B^{\times}$, and integer $r\ge2$, one has

$$\left|\operatorname{N}(B,r,x,a)-\frac{|B^{\times}|^{r-1}}{q}\right|\le r^{\sum_{i=1}^s d_in_i}q^{\frac{(r-1)N-1}{2}}.$$

This conjecture predicts a square-root estimate for the product-trace problem over arbitrary finite semisimple algebras. The paper's introduction states that its purpose is to prove the expected square-root estimate for field extensions and then for arbitrary finite semisimple algebras, but the supplied text does not establish whether this specific conjecture has been resolved.