Hello Viva
Welcome to the SBB Community!
At this time of the year, someone leaving their jacket on the train happens a hundred times a day. It is typival for days with mixed weather, when one needs a jacket in the morning and suddenly the sun is out. There are currently thousands of jackets arriving at SBB’s Lost & Found Center, I have seen that recently with my own eyes.
What concerns the Leman Express, it is important to report the lost item with SBB and SNCF:
https://lostandfound.sbb.ch/home/sbb/SBB/app?Culturename=en
https://www.garesetconnexions.sncf/fr/service-client/a-vos-cotes/objet-perdu-trouve/declaration-perte
Now only speaking about the SBB Lost & Found service: if you have not heard any positive news about your lost property report it does not mean it was not found at all. It can well be the case that not all lost items were yet registered. SBB promised to send status updates and at the time the status update messge was sent, it was not found. Perhaps, it is still in a box waiting for registration. This proces can easily take three weeks, depending on the amount of items found.
Our advice: simply be patient. What helps is a clear description of the jacket (brand, colour, button, etc.) or even better, a photo.
No, there is no need to go to the Cornavin station. All lost and found items on SBB trains are sent to the centralised Lost & Found Service Center in Bern, which cannot be visited.
Kind regards.
Roland