On strong reality of the unipotent Lie-type subgroups over a field of characteristic 2

A group $G$ is called strongly real if its every nonidentity element is strongly real, i.e. conjugate with its inverse by an involution of $G$. We address the classical Lie-type groups of rank $l$, with $l\leqslant4$ and $l\geqslant13$, over an arbitrary field, and the exceptional Lie-type groups over a field $K$ with an element $\eta$ such that the polynomial $X^2+X+\eta$ is irreducible either in $K[X]$ or $K_0[X]$ (in particular, if $K$ is a finite field). The following question is answered for the groups under study: What unipotent subgroups of the Lie-type groups over a field of characteristic 2 are strongly real?