Diffuse BRDF component of the Disney BRDF

Question

In the SIGGRAPH course:

BURLEY B.: Physically Based Shading at Disney, SIGGRAPH 2012 Course: Practical Physically Based Shading in Film and Game Production, 2012.

it is mentioned that some BRDF models include a diffuse Fresnel factor such as:

$$(1-F(\theta_l)) (1-F(\theta_d)).$$

The Disney BRDF itself uses the following diffuse BRDF component (using Sclick's Fresnel approximation):

$$f_d = \frac{\textrm{c_base}}{\pi} (1 + (F_{\textrm{D90}} - 1)(1-\cos\theta_l)^5) (1 + (F_{\textrm{D90}} - 1)(1-\cos\theta_v)^5),$$

where

$$F_{\textrm{D90}} = 0.5 + 2 \text{roughness} \cos^2\theta_d. $$

Where does this come from?

My attempt...

If I evaluate $(1-F(\theta_l)) (1-F(\theta_v))$ (instead of $(1-F(\theta_l)) (1-F(\theta_d))$?) with Schlick's approximation, we get:

$$\left(1-(F_0 + (1-F_0)(1-\cos\theta_l)^5)\right) \left(1-(F_0 + (1-F_0)(1-\cos\theta_v)^5)\right)$$ $$\left(1-F_0 + (F_0-1)(1-\cos\theta_l)^5\right) \left(1-F_0 + (F_0-1)(1-\cos\theta_v)^5\right)$$

If we substitute $F_{\textrm{D90}} = F_0$, we get:

$$\left(1-F_{\textrm{D90}} + (F_{\textrm{D90}}-1)(1-\cos\theta_l)^5\right) \left(1-F_{\textrm{D90}} + (F_{\textrm{D90}}-1)(1-\cos\theta_v)^5\right)$$

This looks similar except for the 2x $-F_{\textrm{D90}}$? Is my reasoning completely wrong or where do I make mistakes? Or am I not aware of some further (common) approximations?