This is not an infectious disease, but probably a soil, root, and/or nutrient issue. According to the link we’ve included at the end of this message, Liriope need good drainage, a slightly acidic soil pH, and regular fertilizer from Feb thru Sept. You’d think, since both plants are in the same area that soil conditions, water content, pH and etc would be the same for both, but we see these situations quite often. It could be that the poor looking plant has a root problem, or a root problem is being caused by the soil properties immediately surrounding this plant. We recommend you refer to the info in this link, specifically the ‘Cultural Practices’ section. One other note – some plants suffer from close proximity to walls and fences because these structures reflect light and heat to the plant, increasing light intensity and temps/heat in excess of the plants tolerance.
http://www.ctahr.hawaii.edu/oc/freepubs/pdf/liriope.pdf
I have a question. I have liriope on either side of the stairs outside and neighborhood cats keep pooping in it. Someone mentioned to me that I could squeeze orange’s on the spot after cleaning it, to detour cats from coming back. Is this going to hurt the plants?