  We demonstrate the equivalence of two different conjectures in the literature
for the holographic entanglement negativity in AdS$_3$/CFT$_2$, modulo certain
constants. These proposals involve certain algebraic sums of bulk geodesics
homologous to specific combinations of subsystems, and the entanglement wedge
cross section (EWCS) backreacted by a cosmic brane for the conical defect
geometry in the bulk gravitational path integral. It is observed that the
former conjectures reproduce the field theory replica technique results in the
large central charge limit whereas the latter involves constants related to the
Markov gap. In this context we establish an alternative construction for the
EWCS of a single interval in a CFT$_2$ at a finite temperature to resolve an
issue for the latter proposal involving the thermal entropy elimination for the
holographic entanglement negativity. Our construction for the EWCS correctly
reproduces the corresponding field theory results modulo the Markov gap
constant in the large central charge limit.
