The linked-pair lower bound for the Casson invariant of positive bireduced diagrams

Let $D$ be a positive bireduced knot diagram. Let $v_2$ be the degree-2 Vassiliev invariant, equivalently the coefficient of $z^2$ in the Conway polynomial, and let $lk(D)$ be the number of linked pairs in $D$.

Linked-pair lower-bound conjecture. One has

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

The claim strengthens the preceding lower bound for positive reduced diagrams under the additional bireducedness assumption. The source gives no proof or resolution for this statement.