Conjecture on the number of segments of each depth

Let $P\subset\mathbb{R}^3$ be a set of $n$ points in convex position. The depth of a segment is the minimum number of points that must be removed from $P$ so that the segment is not contained in the convex hull of the remaining points. Let $s_j(P)$ be the number of segments with depth $j$. The segment-depth conjecture. For $0\leq j\leq \lceil\tfrac{n}{4}\rceil-1$, one has

$$s_j(P)\leq 3n-8j-6.$$

The conjecture is motivated by a construction intended to attain the stated bound throughout the indicated range, and would imply a stronger abundance result for deep pairs in the planar circle problem. It remains open in the source.