Optimal-line intersection conjecture for Chow threefolds

Let $\mathcal{C}\subset\mathbb{P}^3$ be a curve, and consider the associated Chow threefold, secant surface, and tangent curve of lines, together with a data line and an optimal line for the distance problem. Optimal-line intersection conjecture. The optimal line in the Chow threefold always intersects the data line, whereas this does not hold for the secant surface and the tangent curve. The supplied status evidence disproves the universal assertion: the optimal line does not always intersect the data line for the secant surface and tangent curve.