Gromov conjectured in 1985 and LLarull proved in 1998 that: If $g > g_0$ on the sphere, then there exists some point p on the sphere with $Sc(p) < Sc_0(p)$. Here $g, g_0$ are Riemannian metrics and $g_0$ is the standard metric on the sphere. $Sc$ is the scalar curvature. His proof is by contradiction and uses some non-vanishing index on the sphere.
Is the theorem also true for sectional curvature? Or is it the case that if one Riemannian metric is bigger than the other one on the sphere, then there must exist some point such that the scalar curvature at that point is smaller than the other one?
In general, is it true that the bigger the metric, the smaller the curvature (sectional, Ricci or scalar curvaure), at least in the pointwise setting?