  In [Haglund, Remmel, Wilson 2018] the authors state two versions of the so
called Delta conjecture, the rise version and the valley version. Of the
former, they also give a more general statement in which zero labels are also
allowed. In [Qiu, Wilson 2020], the corresponding generalisation of the valley
version is also formulated.
  In [D'Adderio, Iraci, Vanden Wyngaerd 2020], the authors use a pushing
algorithm to prove the generalised version of the shuffle theorem. An extension
of that argument is used in [Iraci, Vanden Wyngaerd 2020] to formulate a valley
version of the (generalised) Delta square conjecture, and to suggest a
symmetric function identity later stated and proved in [D'Adderio, Romero
2020].
  In this paper, we use the pushing algorithm together with the aforementioned
symmetric function identity in order to prove that the valley version of the
Delta conjecture implies the valley version of the generalised Delta
conjecture, which means that they are actually equivalent.
  Combining this with the results in [Iraci, Vanden Wyngaerd 2020], we prove
that the valley version of the Delta conjecture also implies the corresponding
generalised Delta square conjecture.
