A straight line on a Lambert Conformal Conic Projection is for the most part a Great Circle.
A Rhumb Line is a line that crosses the meridians at an equal angle.
A meridian is both a rhumb line (constant angle of 0deg with itself) and a great circle - same centre and radius as the earth. That applies regardless of projection.
What changes with the projection is how the meridians appear on a flat piece of paper.
Regards,
DFC