Let $(W,S)$ be a Coxeter group, $T=\bigcup_{w\in W}wSw^{-1}$ its set of reflections, and $A\subseteq T$. From results of Dyer and Deodhar, we know that the subgroup $W_A$ generated by the elements of $A$ is a Coxeter group, with a canonical system of generators $S_A$.

In the final remarks to his note, Deodhar points out that it would be interesting to compare the Bruhat order on $(W_A,S_A)$ to the restriction of the Bruhat order of $(W,S)$ to $W_A$, and that he plans to take up this project.

**My question is:** Has this been done? I would be happy with any references, but I am particularly interested in the following special case: Assume that $\beta_1,\ldots, \beta_k$ are positive roots in $W$'s root system. Is it possible to relate elements $w\in W$ satisfying $w\lessdot wr_{\beta_i}$ for all $i=1,\ldots ,k$, where $\lessdot$ denotes covering in the Bruhat order of $(W,S)$ and $r_{\beta_i}$ is the reflection across $\beta_i$ to elements $\widetilde{w}\in W_A$ with the same covering relations, but now in the Bruhat order of $(W_A,S_A)$?

**An example:** Consider the root system of $G_2$ with $\alpha$ the short, and $\beta$ the long simple root. The only element of $W$ satisfying $w\lessdot wr_\alpha, wr_{3\alpha+\beta}$ is $w=r_\alpha r_\beta r_\alpha r_\beta$, so it is true that $wr_\beta <w$, $wr_{\alpha+\beta} <w$, $wr_{3\alpha+2\beta}<w$, $wr_{2\alpha+\beta}<w$. Now let $\alpha,\beta$ be positive roots forming a sub-root system of type $G_2$ in some large root system. Do the four relations above still hold for the Bruhat order of the large group?