Answer
The cone will slide first.
Work Step by Step
The center of mass of a cone is 1/4 of its height from the base. We know that the cone will top when the center of mass is over the corner of the cone, which is h/6 away from its center. Thus:
$\theta = tan^{-1}(\frac{h/6}{h/4})=33.69^{\circ}$
We see when the cone will slide:
$\theta = tan^{-1}(.63)=32.21^{\circ}$
The cone will slide first.
