Let $(W,S)$ be a Coxeter group. Then the Kazhdan-Lusztig polynomials $P_{x,w}$ are known to satisfy the following recursive identity:

For $x < w$ and $s \in S$ satisfying $\ell(sw) < \ell(w)$, define

$$c = c_s(x) := \begin{cases} 0, \,\,\,\text{ if }x < sx\\ 1, \,\,\,\text{ if }x > sx. \end{cases}$$

Then, $P_{x,w}$ satisfies $$P_{x,w} = q^{c−1}P_{sx,sw} + q^c P_{x,sw} −\sum_{sz<z<sw} \mu(z, sw)q^{(\ell(z)-\ell(w))/2} P_{x,z},$$ where $\mu(z, sw)$ is the coefficient of $q^{(\ell(sw)−\ell(z)−1)/2}$ in $P_{z,sw}$.

Is there a known identity similar to the above for the inverse Kazhdan-Lusztig polynomials $Q_{x,w}$?

In the finite case, we have $Q_{x,w}=P_{w_0w,w_0x}$ where $w_0\in W$ is the longest length element, but I am interested in the affine or general Coxeter case.