Spanning conjecture for TD systems

Let ${\cal F}$ be a field, let $V$ be a finite-dimensional nonzero vector space over ${\cal F}$, and let $\Phi$ be a TD system on $V$ with diameter $d$. Let $R$ and $L$ be the maps from Definition

, let $U_0$ be the space from Definition

, and choose any nonzero vector $v\in U_0$. Spanning conjecture for TD systems. $V$ is spanned by the vectors

$$L^{i_1}R^{i_2}L^{i_3}R^{i_4}\cdots R^{i_n}v,$$

where $n$ is nonnegative and even, and $i_1,i_2,\ldots,i_n$ are integers satisfying $0\leq i_1<i_2<\cdots<i_n\leq d$. For $d=3$, these vectors are $v$, $Rv$, $R^2v$, $R^3v$, $LR^2v$, $LR^3v$, $L^2R^3v$, and $RL^2R^3v$. The conjecture has been proved for $d\leq 3$ by Tanabe and implies the subconstituent-dimension bound conjecture; the general case remains open.