This is a follow-up question to Reference request: Presentations of Hecke algebras. In the common definition $$H_t = \langle T_1,\dotsc, T_n \mid \text{braid relations, and for all i: $T_i^2 = t + (t-1)T_i$}\rangle$$ of the Hecke algebra with KL-basis $C'_w$ and KL-polynomials $n_{vw}$, the KL-conjecture states that the multiplicity of the simple module $L(w)$ in a Jordan-Hölder series for the Verma module $M(v)$ is given by $$[M(v):L(w)]=n_{w_0v,w_0w}.$$ There is also an interpretation of the coefficients of different exponents, due to Jantzen. However, owing to the lectures I attended, I am more used to the presentation $$H'_q = \langle H_1,\dotsc, H_n \mid \text{braid relations, and for all i: $H_i^2 = 1 + (q^{-1}-q)H_i$}\rangle.$$ For $q=t^{-1/2}$ and $H_s=qT_s$ both are isomorphic. Let $p_{vw}$ be the base change coefficients from the basis $H_\bullet$ to $C'_\bullet$.
Question: Is it true that $p_{vw}=n_{w_0w,w_0v}$?
Attempt: We know $C'_{w}=q^{-\ell (w)/2}\sum _{{v\leq w}}n_{vw}T_{v}$. By putting $T_v=t^{\ell(v)/2}H_v$ into the formula, we get $$C'_{w}=\sum _{{v\leq w}}\underbrace{q^{\ell{v}/2-\ell (w)/2} n_{vw}}_{p_{vw}} H_{v}$$ I don't see how to prove the requested property for $p_{vw}$.