Material for joint sealing stairs/wall, sound insulation?

Good day, we recently moved into a house built in 2012, which has three enclosed concrete staircases connecting the floors. We have now noticed the extreme noise transmission in the house, especially between the floors despite concrete ceilings and enclosed staircases. This is because the staircases are not properly connected to the wall; on the lower floor, you can even see light through the gaps on the sides into the floor above. My question: how do we best seal these cavities? My father suggested construction foam, but does it also sufficiently block noise? Once you spray something into the gap, you can't remove it anymore..... I would be grateful for tips on which material should be used for sealing. Silicone is presumably not an option, as the gaps are several centimeters wide in some places. Thank you very much for an assessment.

the joints are several centimeters wide in places

Could it be that this is deliberately left open in order to achieve a necessary decoupling of the staircase from the wall to prevent impact sound transmission?

Could it be that this is deliberately left open in order to achieve a necessary decoupling of the staircase from the wall to prevent impact sound transmission?

That will be the case. If the stairwell is open anyway, sealing the joints won’t help. Expanding foam is the wrong material. Put simply, you insert a thick cord into the joints and seal over it with a permanently elastic material, silicone, or bituminous, or...

The stairs are closed, but purely in terms of noise transmission from floor to floor, it sounds as if they were open... For example: a baby cries in the attic, you can hear it in the lower floor as if you were standing right next to it, despite concrete ceilings and sealed doors. We have clearly identified the stairs as the culprits. I am attaching a photo to clarify which joint I mean. As you can see, the joint is also uneven. Impact noise is not a problem; the stairs are extremely quiet to walk on.

yes, insert a cord and then properly apply silicone.
I would doubt that the stairs are to blame, but a try only costs a few euros.

And this crack is supposed to make more of a difference than the staircase itself? Hard for me to believe. Silicone or, if it's really large, maybe a [Kompriband]. Not expanding foam.