The constructive recognition complexity conjecture for special linear groups

Let $d\geq 4$, let $q$ be a prime power, and let $\operatorname{SL}(d,q)$ be given by a generating set. Let $\xi$ denote an upper bound on the number of field operations for computing a random element in $\operatorname{SL}(d,q)$, and let $\zeta(q)$ denote an upper bound on the number of field operations for constructively recognising $\operatorname{SL}(4,q)$. The algorithm $\operatorname{StandardGenerators}$ is the one-sided Monte Carlo algorithm from the paper for constructive recognition of $\operatorname{SL}(d,q)$. The complexity conjecture. The complexity of $\operatorname{StandardGenerators}$ is

$$\mathcal{O}\left(d^3 \log(d) + d^2 \log(d) \log(\log(d)) \log(q) + \frac{\log(d)}{\log(\log(d))}\xi + \zeta(q)\right).$$

This would complete the complexity analysis of the constructive recognition algorithm; the paper states that the detailed analysis is almost complete, while the corresponding adaptations to other classical groups are deferred to follow-up work.