Sewing machine kind of noise only when no power from engine to wheels

Hello to all C3 fans.
I need some advice/oppinions about the following strange (for me) noise on my 2002 1.1 Liter C3 (210 000 Km ~ 130000 miles).
When driving and the power is from engine to the wheels everything is NORMAL, no abnormal noise!
When going down hill (or even just rolling without using the gas pedal) a sewing machine kind of noise shows, and the frequency increases the faster the car goes.
The noise can be heard no matter if the gearbox is in neutral or ingear (any gear). Also can be heard no matter if the clutch is in or out.

And, more strange in my oppinion, if I'm driving backwards, using reverse gear or just let the car roll, NO noise shows.

Citroen local shop said the gearbox have a problem... but they didn't look too sure about that. And the price they want to charge to replace/fix the gearbox ...
Should I send the car to the scrapyard?
Please forgive me for any mistakes and if the post sounds confusing, but English is not my first language.
Thanks in advance for any tips/ideas

Hi,

I don't know the answer. A second hand gearbox and doing the work yourself can be cheaper than paying Citroen

Would it be possible to record a YouTube video of the sound you get, then attach it here so we can hear the sound too.

Here is a short video where (I think) you can hear the mentioned noise:
https://youtu.be/sZgV2uU228Q

Gearboxes are not my forte but apparently the input shaft bearing fails quite often. A good mechanic should be able to fix it for a reasonable price but wether that's economical is up to you.

Today I have tried one more thing (I have not tried before because is not normal to drive with the engine not running)
Well even with the engine not running, but driving down hill the noise can clearly be heard, no matter if the clutch is in or out.
I am really confused. The funny part is Citroen people didn't test the car with engine stopped.