Adding a suitable unknot to any link equates bridge number and meridional rank

Given any link upper L subset of or equal to upper S cubed L⊆ S³ L ⊆ S 3 , we show that it is possible to embed an unknot U in its complement so that the link upper L union upper U L∪ U L ∪ U satisfies the Meridional Rank Conjecture (MRC). The bridge numbers in our construction fit into the equality beta left parenthesis upper L union upper U right parenthesis equals 2 beta left parenthesis upper L right parenthesis minus 1 equals rank left parenthesis pi 1 left parenthesis upper S cubed minus left parenthesis upper L union upper U right parenthesis right parenthesis right parenthesis β(L∪ U)=2β(L)-1=rank(π₁(S³ (L∪ U))) β ( L ∪ U ) = 2 β ( L ) − 1 = rank ( π 1 ( S 3 ∖ ( L ∪ U ) ) ) . In addition, we prove the MRC for new infinite families of links and distinguish them from previously settled cases through an application of bridge distance.