  We prove the Schr\"oder case, i.e. the case $\langle \cdot,e_{n-d}h_d
\rangle$, of the conjecture of Haglund, Remmel and Wilson (Haglund et al. 2018)
for $\Delta_{h_m}\Delta_{e_{n-k-1}}'e_n$ in terms of decorated partially
labelled Dyck paths, which we call \emph{generalized Delta conjecture}. This
result extends the Schr\"oder case of the Delta conjecture proved in
(D'Adderio, Vanden Wyngaerd 2017), which in turn generalized the
$q,t$-Schr\"oder of Haglund (Haglund 2004). The proof gives a recursion for
these polynomials that extends the ones known for the aforementioned special
cases. Also, we give another combinatorial interpretation of the same
polynomial in terms of a new bounce statistic. Moreover, we give two more
interpretations of the same polynomial in terms of doubly decorated
parallelogram polyominoes, extending some of the results in (D'Adderio, Iraci
2017), which in turn extended results in (Aval et al. 2014). Also, we provide
combinatorial bijections explaining some of the equivalences among these
interpretations.
