Let $K$ be a convex body in $\mathbb{R}^3$. Suppose $K$ is held at some position and orientation on an inclined plane, and released. Let there be sufficient friction so that it rolls without slippage. My question is:

Q. If $K$ rolls along a straight line, i.e., if the point of contact along the inclined plane is a single straight line, what can we conclude about the shape of $K$? In other words, which $K$, when properly oriented, roll straight?

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(Figure from Which convex bodies roll along closed geodesics?)}

It seems that if $K$ is a smooth surface of revolution about an axis $X$, and $K$ has reflective symmetry about a plane orthogonal to $X$ (as in the above illustration), then $K$ rolls straight. But perhaps a wider class of bodies also roll straight. Perhaps reflective symmetry is not necessary; perhaps equal moments of inertia about $X$ in the two halves suffice? Or would any asymmetry cause a wobble in the footprint?

I would be interested in learning of any class of shapes that roll straight, especially non-symmetric shapes.