In this post on the consecutive integers $b^2,2a^2,3c^2$, I asked whether the trivial solution $a=b=c=1$ was the only one. At this time, that question appears to have been answered in the affirmative (proven by @san).
Now my question is a generalization, which may or may not already have an answer (cf. Erdős's study of consecutive integers and their factors, powerful numbers, etc.):
QUESTION: For which triples of square-free integers $(r,s,t)$ do triples of integers $(x,y,z)$ exist such that $rx^2$, $sy^2$, and $tz^2$ are consecutive integers?
Example: The question referenced at the top of this post is the special case $(r,s,t)=(1,2,3)$, for which the only solution triple $(x,y,z)$ is the trivial solution $(x,y,z)=(1,1,1)$, leading to the consecutive integers $$(rx^2,sy^2,tz^2)=(1,2,3).$$
Has this problem already been tackled? Obviously there are many tools by which it can be attacked (e.g., complete solutions to the equation $uX^2+vY^2+wZ^2=0$).