Step 1: Recall WHO guidance on timing.
The recommended gap before giving a semen sample is short, in the range of about $2$ to $7$ days.
Step 2: See what six weeks does.
Six weeks is well past that window. Holding sperm in the epididymis for so long lets more of them age and die or lose motility, so the sample actually gets worse, not better. That makes the six week claim false, which is the answer we want since the question asks for the one FALSE statement.
Step 3: Confirm motility as a quality marker.
Motility, how well sperm swim forward, is one of the main things checked in a semen analysis because it tracks directly with the chance of natural fertilisation. So this claim is true.
Step 4: Confirm the collection site rule.
Because sperm lose motility quickly once outside the body, the sample should ideally be produced right at the lab, or delivered within a short set time if produced elsewhere. So this claim is true.
Step 5: Confirm the fructose fact.
Fructose is made by the seminal vesicles and is normally present in semen. No fructose points to a blocked ejaculatory duct or absent seminal vesicles. This claim is true too.
Step 6: Conclude.
Only the six week abstinence claim is false.
\[ \boxed{\text{Abstinence for 6 weeks provides the best sample}} \]