Ingleton score is at most one

Statement
Suppose $$G$$ is a finite group and $$G_1,G_2,G_3,G_4$$ are all subgroups (possibly equal, possibly distinct) of $$G$$. For any subset $$\alpha$$ of $$\{ 1,2,3,4 \}$$, denote by $$G_\alpha$$ the subgroup $$\bigcap_{i \in \alpha} G_i$$. For convenience, we will write $$\alpha$$ simply as a concatenated string of its elements, so for instance, $$G_{134}$$ stands for $$G_{\{ 1,3,4 \}}$$ and is defined as $$G_1 \cap G_3 \cap G_4$$.

Recall that the Ingleton score $$s$$ of this tuple is defined as follows, where the base of the logarithm is the same for both the numerator and the denominator:

$$s = \frac{\log r}{\log |G/G_{1234}|}$$

where $$r$$, also called the Ingleton ratio, is defined as:

$$r = \frac{|G_{12}||G_{13}||G_{14}||G_{23}||G_{24}|}{|G_1||G_2||G_{34}||G_{123}||G_{124}|}$$

Alternatively, $$r$$ can be defined as:

$$r = \frac{|G_{14}G_{24}||G_{13}G_{23}|}{|G_1G_2||G_{34}|}$$

We will show that $$s \le 1$$.

Proof
We will use the second definition of $$r$$.