Projection-diameter conjecture for space curves

Let $\gamma$ be a curve in $\mathbb R^n$ of length $L$. An orthogonal projection is the projection onto an $(n-1)$-dimensional linear subspace.

Projection-diameter conjecture. There is some orthogonal projection to $\mathbb R^{n-1}$ in which the image of $\gamma$ has diameter at most

$$\frac{L}{\pi}.$$

The claim would extend the planar projection argument to curves in higher dimensions and would imply that every such curve has a pair of opposite points with distortion at least $[?]$; the source gives no resolution of the conjecture.