The Petrie conjecture asserts that if a homotopy
C
P
n
\mathbb {CP}^n
admits a non-trivial circle action, its Pontryagin class agrees with that of
C
P
n
\mathbb {CP}^n
. Petrie proved this conjecture in the case where the manifold admits a
T
n
T^n
-action. An almost complex torus manifold is a
2
n
2n
-dimensional compact connected almost complex manifold equipped with an effective
T
n
T^n
-action that has fixed points. For an almost complex torus manifold, there exists a graph that encodes information about the weights at the fixed points. We prove that if a
2
n
2n
-dimensional almost complex torus manifold
M
M
only shares the Euler number with the complex projective space
C
P
n
\mathbb {CP}^n
, the graph of
M
M
agrees with the graph of a linear
T
n
T^n
-action on
C
P
n
\mathbb {CP}^n
. Consequently,
M
M
has the same weights at the fixed points, Chern numbers, cobordism class, Hirzebruch
χ
y
\chi _y
-genus, Todd genus, and signature as
C
P
n
\mathbb {CP}^n
, endowed with the standard linear action. Furthermore, if
M
M
is equivariantly formal, the equivariant cohomology and the Chern classes of
M
M
and
C
P
n
\mathbb {CP}^n
also agree.