Why postgresql trigger doesn't launch?

I use postgresql 11. I think my problem comes from a trigger that doesn't do an update so the next trigger doesn't launch.

I have a table projet with columns : projet_temps_doe, projet_temps_etudes, projet_temps_globale.

The goal is to update each columns depending on other columns values. The idea is : projet_temps_globale = projet_temps_doe + projet_temps_etudes.

I have a first trigger on projet_temps_doe which works perfectly:

create function temps_globale_doe() returns trigger
    language plpgsql
as
$$
begin
        new.projet_temps_globale_doe :=  new.projet_temps_doe_gc_bts + new.projet_temps_doe_gc_nra;
        return new;
    end;
$$;

CREATE TRIGGER temps_globale_doe
    BEFORE UPDATE OF projet_temps_doe_gc_bts, projet_temps_doe_gc_nra
    ON public.projet
    FOR EACH ROW
    EXECUTE PROCEDURE public.temps_globale_doe();

I have a similar trigger on projet_temps_etudes which works perfectly too.

Then the trigger I struggle with on projet_temps_globale :

create trigger maj_temps_globale_projet
    before update of projet_temps_doe, projet_temps_etudes on projet
    for each row
    execute procedure maj_temps_globale_projet();

create or replace function maj_temps_globale_projet()returns trigger
language plpgsql
as
    $$
    begin
        new.projet_temps_globale := new.projet_temps_doe + new.projet_temps_etudes;
        raise info 'TEST!!';
        return new;
    end;
    $$;

When projet_temps_doe and/or projet_temps_etudes are updated via triggers my last trigger doesn't launch. However when I manually change projet_temps_doe and/or projet_temps_etudes values the trigger maj_temps_globale_projet is fired.

I want to learn from this, so, if possible, explain to me what I'm doing wrong here, or if my approach is lacking insight.