Jenny, I don't know what the Mrs will say to that question but I can relate a convo I had with my H's sister one day. She was griping about how her husband wanted sex all the time and I presented a scenario like what you said to her. She looked shocked for a moment and said, "Well I would start pushing for sex if he suddenly didn't want it anymore." I asked why and she said that she wouldn't feel attractive if he didn't want her.
Then one time about a month ago, I was trying to prop up my H's esteem during a talk we were having and I said, "Look H, I know that in another relationship you'd be the high drive partner and I feel bad that we are having these problems that you wouldn't be having with another woman." He replied, Oh that's not true. If she never wanted sex, I wouldn't either. I'd just go along with whatever my wife wanted, probably.
(btw, NOP, this is another example of what I mean by sexually passive. He is not passive in any other area of our life, but he will "go along" with whatever his wife wants, even if it were near celibacy? With absolutely no motivation to go after his own needs? How attractive and sexy!)
Anyway, my point is that I think it would depend on the particular LD person, on how they would react.