The linked-pair lower bound for the second Vassiliev invariant

Let $D$ be a positive bireduced knot diagram, and let $lk(D)$ denote the number of linked pairs in $D$. Let $v_2$ be the second Vassiliev invariant. The linked-pair lower-bound conjecture.

$$v_2\geq \frac{lk(D)}{4}.$$

This conjecture seeks a crossing-combinatorial lower bound for $v_2$ on positive diagrams; the supplied text gives no evidence of a resolution.