  The Shuffle Theorem of Carlsson and Mellit gives a combinatorial expression
for the bigraded Frobenius characteristic of the ring of diagonal harmonics,
and the Delta Conjecture of Haglund, Remmel and the second author provides two
generalizations of the Shuffle Theorem to the delta operator expression
$\Delta'_{e_k} e_n$. Haglund et al. also propose the Extended Delta Conjecture
for the delta operator expression $\Delta'_{e_k} \Delta_{h_r}e_n$, which is
analogous to the rise version of the Delta Conjecture. Recently, D'Adderio,
Iraci and Wyngaerd proved the rise version of the Extended Delta Conjecture at
the case when $t=0$. In this paper, we propose a new valley version of the
Extended Delta Conjecture. Then, we work on the combinatorics of extended
ordered multiset partitions to prove that the two conjectures for
$\Delta'_{e_k} \Delta_{h_r}e_n$ are equivalent when $t$ or $q$ equals 0, thus
proving the valley version of the Extended Delta Conjecture when $t$ or $q$
equals 0.
