So, you're showing that for any root $\beta$, $s_\beta$ can be written with a symmetric word of length $2\mathrm{dp}(\beta) -1$, right? Then, to answer the question/claim in my post, you'd essentially combine that with the equality written there. I'm not sure of what you assume and what you conclude. (And thanks!)

@JimHumphreys: I'm not sure I understand the remark; is some sort of rewriting of the question in order?

@MattSamuel: I'd be happy with both!