Assume $0 < p_1 \le p_2\le p_3 \le p_4$. What is the maximum of the following expression?
$ \frac{\left(p_1+p_4\right)\left(p_2+p_3\right)}{\left(p_1+p_3\right)\left(p_2+p_4\right)} $
Is that bounded by a constant?
Assume $0 < p_1 \le p_2\le p_3 \le p_4$. What is the maximum of the following expression?
$ \frac{\left(p_1+p_4\right)\left(p_2+p_3\right)}{\left(p_1+p_3\right)\left(p_2+p_4\right)} $
Is that bounded by a constant?
$\frac{\left(p_1+p_4\right)\left(p_2+p_3\right)}{\left(p_1+p_3\right)\left(p_2+p_4\right)} = 1 + \frac{\left(p_2-p_1\right)}{\left(p_1+p_3\right)} \times \frac{\left(p_4-p_3\right)}{\left(p_2+p_4\right)} $ and the right hand side is greater than or equal to $1 + 0 \times 0$ but less than $1 + 1 \times 1$.
The values $(1,n,n,n^2)$ achieve the lower bound when $n=1$ but approach the upper bound when $n$ increases without limit: $n=400$ gives a figure over $1.99$.