The Vogan-chain characterization conjecture for orbital-variety closures

Let ${\bf T}_n$ be the set of standard Young tableaux with $n$ boxes, let $I_T$ be the primitive ideal attached to $T$, let ${\mathcal V}_T$ be its orbital variety, and let $\stackrel{\rm V-Ch}{\leq}$ denote the Vogan-chain order. Vogan-chain characterization conjecture. For any $T,S\in{\bf T}_n$,

$$\overline{\mathcal V}_T\subset\overline{\mathcal V}_S$$

if and only if

$$I_T\supset I_S,$$

and this happens if and only if $S\stackrel{\rm V-Ch}{\leq}T$. The computations in the paper support this as the main conjectural identification of geometric, algebraic and Vogan-chain orders.