Let
G
G
be a connected and simply connected nilpotent Lie group,
K
K
an analytic subgroup of
G
G
and
π
\pi
an irreducible unitary representation of
G
G
whose coadjoint orbit of
G
G
is denoted by
Ω
(
π
)
\Omega (\pi )
. Let
U
(
g
)
\mathscr U(\mathfrak g)
be the enveloping algebra of
g
C
{\mathfrak g}_{\mathbb C}
,
g
\mathfrak g
designating the Lie algebra of
G
G
. We consider the algebra
D
π
(
G
)
K
≃
(
U
(
g
)
/
ker
(
π
)
)
K
D_{\pi }(G)^K \simeq \left (\mathscr U(\mathfrak g)/\operatorname {ker}(\pi )\right )^K
of the
K
K
-invariant elements of
U
(
g
)
/
ker
(
π
)
\mathscr U(\mathfrak g)/\operatorname {ker}(\pi )
. It turns out that this algebra is commutative if and only if the restriction
π
|
K
\pi |_K
of
π
\pi
to
K
K
has finite multiplicities (cf. Baklouti and Fujiwara [J. Math. Pures Appl. (9) 83 (2004), pp. 137-161]). In this article we suppose this eventuality and we provide a proof of the polynomial conjecture asserting that
D
π
(
G
)
K
D_{\pi }(G)^K
is isomorphic to the algebra
C
[
Ω
(
π
)
]
K
\mathbb C[\Omega (\pi )]^K
of
K
K
-invariant polynomial functions on
Ω
(
π
)
\Omega (\pi )
. The conjecture was partially solved in our previous works (Baklouti, Fujiwara, and Ludwig [Bull. Sci. Math. 129 (2005), pp. 187-209]; J. Lie Theory 29 (2019), pp. 311-341).