Individual Cases each have three abilities. The top ability is always active. The middle ability, preceded by "to solve," sets out the conditions under which you can solve the Case. At the beginning of your end step, if the Case is unsolved and if you've met the condition described in the middle ability, the Case becomes solved. The described condition must be true both as your end step begins and as the "to solve" ability tries to resolve. For Case of the Filched Falcon, if you control three or more artifacts as your end step begins, the "to solve" ability will trigger and go on the stack. If you still control three or more artifacts as that ability tries to resolve, the Case will be solved.
Combined with the example card used in the article, we can determine that the check is made at the time of the to-solve end step trigger. So, you have to meet the condition on your turn. You can only solve Case of the Ransacked Lab if it's your turn and you cast four of those kinds of spells before your end step.