We show that, for any
n
≥
2
n\geq 2
, there exists a homogeneous space of dimension
d
=
8
n
−
4
d=8n-4
with metrics of
R
i
c
d
2
−
5
>
0
Ric_{\frac {d}{2}-5}>0
if
n
≠
3
n\neq 3
and
R
i
c
6
>
0
Ric_6>0
if
n
=
3
n=3
which evolve under the Ricci flow to metrics whose Ricci tensor is not
(
d
−
4
)
(d-4)
-positive. Consequently, Ricci flow does not preserve a range of curvature conditions that interpolate between positive sectional and positive scalar curvature. This extends a theorem of Böhm and Wilking [Geom. Funct. Anal. 17 (2007), pp. 665–681] in the case of
n
=
2
n=2
.