Let
R
=
k
[
x
,
y
]
R=k[x,y]
be a polynomial ring over a field
k
k
of prime characteristic
p
p
and let
E
E
denote the injective hull of
k
k
(which is isomorphic to
H
(
x
,
y
)
2
(
R
)
H^2_{(x,y)}(R)
). We prove that
E
E
is not an injective object in the category of graded
F
F
-modules over
R
R
. This answers in the negative a question raised by Lyubeznik-Singh-Walther [J. Eur. Math. Soc. (JEMS) 18 (2016), pp. 2545- 2578].