On a Question of Glasby, Praeger, and Xia in Characteristic $2$

Recently, Glasby, Praeger, and Xia asked for necessary and sufficient conditions for the `Jordan partition' $\lambda(m,n,p)$ to be standard. Previously we gave such conditions when $p$ is any odd prime. Here we give such conditions when $p=2$. Our main result is that $\lambda(m,n,2)$ is never standard for $4 \leq m \leq n$.