In the title you can see the question for the case $n = 4$, which was my original doubt. I couldn't prove that, couldn't even start on how to think about it.
Also, it would be nice to know if that can be generalized to a case of $n$ integers, that is:
What are the minimal conditions for the following formula to hold true? $$ LCM\left(\displaystyle\frac{\prod_{i=1}^{n} x_{i}}{x_1},\frac{\prod_{i=1}^{n} x_{i}}{x_2},\cdots,\displaystyle\frac{\prod_{i=1}^{n} x_{i}}{x_n}\right)=\displaystyle\prod_{i=1}^{n} x_{i},\text{ with }x_i\in\mathbb{Z}^+. $$
Take any prime $p$, and let it occur to powers $w,x,y,z$ in $a,b,c,d$ respectively. Then $p$ occurs to power $v$ in the LCM, where $$v=\max(x+y+z,w+y+z,w+x+z,w+x+y)$$
And it occurs to power $w+x+y+z$ in $abcd$. So the condition is that for all primes $p$, $$\max(x+y+z,w+y+z,w+x+z,w+x+y)=w+x+y+z$$ which is true if and only if at least one of $w,x,y,z$ is zero.
Putting it another way: the condition is that no prime occurs as a factor of all four values $a,b,c,d$.
And this generalises to $n$ terms in the obvious way.