Let $K$ be a solid, homogenous convex body in $\mathbb{R}^3$. Place $K$ on an inclined plane, and let it roll down the plane, under some reasonable assumptions of friction between $K$ and the plane, and tilt angle of the plane, so that it rolls without slippage under the influence of gravity. Let the $y$-axis represent straight down the gradient of the plane, and $x$ the distance of the contact point(s) to the left or right of straight-down.

Q. Does there exist some $K$ so that, from a particular starting position, $K$ rolls such that the sideways displacement $x$ grows without bound the further down $y$ it rolls?

For example, perhaps there is a $K$ so that this (entirely fanciful) sideways "crab-walking" path is the trace of the point of contact:
         
Or perhaps no such $K$ exists—It is mathematically impossible? I neither have strong intuitions here, nor a clear idea on how to settle the question one way or the other. I have no particular candidate $K$, although this asymmetric oloid-like object is what piqued my interest: