Consider a curve $\gamma_s= U_s^{\dagger} b U_s$ in $\mathfrak{su}(n)$ where $U_s$ is a smooth curve on $SU(n)$ (starting at $U_0 = \mathbb{I}$) and nonzero $b\in \mathfrak{su}(n)$ and $s \in[0,T]$ for some $T>0$.
assume that the span of all the points on $\gamma_s$ has dim one less than $\mathfrak{su}(n)$, i.e. a co-rank 1 singular curve.
Is it now true that either of the `surfaces' $Q_2(x,y):= \gamma_x \gamma_y$ or $Q_3(x,y,z):= \gamma_x \gamma_y \gamma_z$, have the property that the span of the points on the surface includes all of $\mathfrak{su}(n)$ for $x,y,(z) \in [0,T]$?