Lower-central-series quotient conjecture for the group G

Let $G$ be the group considered in the paper, let $H_i$ be the associated upper-triangular subgroups, and let $\lambda_i(G)$ and $\gamma_i(G)$ denote the terms of the lower exponent-$2$ series and the lower central series of $G$, respectively. Lower-central-series quotient conjecture. For $i\geq 1$ and $i\geq 2$, respectively, one has

$$\lambda_i(G)=G\cap H_i,$$

and

$$\lambda_i(G)/\lambda_{i+1}(G)\cong\gamma_i(G)/\gamma_{i+1}(G).$$

The authors state this as a conjecture based on MAGMA calculations. If true, it would transfer the established width and periodicity results from the lower exponent-$2$ series to the lower central series; no resolution is supplied in the given text.