When a glider or unpropelled aircraft experiences headwind, I understand that the flight velocity and sink velocity for the "new" maximum possible range both increase (see velocity polar from How does wind affect the airspeed that I should fly for maximum range in an airplane?).
Intuitively I would think this means that $C_{L}$ needs to decrease so that the now higher $\frac{1}{2}\rho v^2$ can be compensated (small glide slope assumption $L \approx G$ i.e. $L >> D$).
For the theoretical $C_{L_{R_{max}}}$ equation for a symmetric $C_{L}-C_{D}$ polar, I don't see this playing a role:
$C_{L_{R_{max}}}=\sqrt{C_{D_0}\pi\Lambda e} $
...as $C_{D_0}$ to my knowledge doesn't change with velocity.
Does lift coefficient $C_{L_{R_{max}}}$ for maximum range change for a glider experiencing headwind?