Cascade updates on tables and pivot tables with laravel and Eloquent - eloquent

Question

I created a model nammed sector. For this model i created methods to make links between the database table :
A sector has many valuechain with many segments, which have many
keyneeds
A valuechain has many segments
A segment has many keyneeds ...
It has been translated this way :
public function valuechains()
{
return $this->hasMany('App\Valuechain');
}
public function segments()
{
return $this->hasManyThrough('App\Segment', 'App\Valuechain');
}
public function keyneeds()
{
return $this->hasManyThrough('App\Keyneed', 'App\Segment', 'App\Valuechain');
}
When i soft delete a sector I want to create a cascade on other tables and make updates.
My destroy method is :
public function destroy($id)
{
$sector = Sector::findOrFail($id);
// on update lang_sector pour chaque id
$sector_ids = $sector->langs()->allRelatedIds();
foreach ($sector_ids as $id){
$sector->langs()->updateExistingPivot($id, ['lang_sector.deleted_at' => Carbon::now()]);
}
$sector->valuechains()->update(
[
'valuechains.deleted_at' => Carbon::now(),
'valuechains.updated_at' => Carbon::now(),
]
);
$sector->segments()->update(
[
'segments.deleted_at' => Carbon::now(),
'segments.updated_at' => Carbon::now(),
]
);
$sector->keyneeds()->update(
[
'keyneeds.deleted_at' => Carbon::now(),
'keyneeds.updated_at' => Carbon::now()
]
);
Sector::where('id', $id)->delete();
return redirect()->route('sectors.index')->with('success', 'Sector deleted');
}
I have pivot table for valuechains, segments and keyneeds and i also want to update the updated_at and deleted_at columns on those tables...
For those 3 tables i added to the model the lang method for defining the pivot relation and defining the fields i have inside the pivot table:
public function langs() {
return $this->belongsToMany('App\Lang')
->withPivot(
'vcname',
'vcshortname',
'vcdescription',
'vcshortdescription',
'created_at',
'updated_at',
'deleted_at'
);
}
I have an error message :
SQLSTATE[23000]: Integrity constraint violation: 1052 Champ: 'updated_at' dans field list est ambigu (SQL: update segments inner join valuechains on valuechains.id = segments.valuechain_id set valuechains.deleted_at = 2018-05-09 07:34:30, valuechains.updated_at = 2018-05-09 07:34:30, segments.deleted_at = 2018-05-09 07:34:30, segments.updated_at = 2018-05-09 07:34:30, **updated_at** = 2018-05-09 07:34:30 where valuechains.sector_id = 2)