The dominance conjecture for flag h-vector symmetry

Let $r$ be a positive integer, let $\beta$ be the order-reversing permutation in $S_{r+1}$ defined by $\beta(i)=r-i+2$, and let $T$ be a subset of $[r]$ such that

$$T\subseteq T\circ\beta,$$

where

$$T\circ\beta=\{i\in[r]:r-i+1\notin T\}.$$

Dominance conjecture. If $T\subseteq T\circ\beta$, then $T\circ\beta$ dominates $T$.

If true, this would provide a family of dominance relations implying inequalities between the corresponding flag $h$-vector entries for every geometric lattice of sufficiently large rank. The supplied text gives an example with $T=\{1\}\subseteq[3]$, but does not state a proof or resolution of the general claim.