Audio Note speakers

AN-Js are great speakers I had a pair until recently and they should play just fine with your LTA amp. Regarding cone types my dealer recommends paper and I went with paper he says it gives you fuller bass vs the hemp cones. My pair were wired with Lexus copper and sounded fantastic. The only reason I moved on was for my dream horn speakers. 

I really like the AN-J speakers too.  As Jond mentioned, the paper cone speaker is slightly warmer sounding, but the difference is not that big and both sound very good.  I don't know about your specific LTA amp, but the ones I've heard are slightly on the leaner side compared to Audio Note amps so it might make sense to stick with copper speaker cables, particularly at any reasonable price level.  I like the Audio Note cables but I've also heard the AN-J's with Acrolink copper cables and the sound was very good.  

Good friend runs AN-J’s as well, he drives them with AudioNote’s Cobra amp. That is a wonderfully musical system. Great synergy. All the more expensive versions of AN’s speakers run the Hemp drivers for a reason they are more detailed and musical from my experience. Get the best ones you can afford. I would also consider the AN Cobra amp as another alternative to the LTA, the synergy is there with he two. The Cobra-AN-J is a combo I could easily live with forever. Oh and the DAC in the Cobra is surprisingly good.

I owned the An-J’s for 2 years, and had both paper and hemp. Martin Grennall at AN advised the hemp because he feels it’s richer sounding. I think if you prioritize tonal richness and fluidity, hemp is the correct choice. If you prefer dynamic articulation, paper works better. 
I sold my J’s and upgraded to the E’s, which provide significantly more satisfaction in my space. I’m not sure of the size of your room, but the body of the music, the bass, and the power all increase in quality and size with the E’s. They work remarkably well in small rooms, and I cherish their ability to fill my space effortlessly with a wall-to-wall hologram of sound vs the J’s more dainty projection. It’s a difference experience, and which is better is subjective. 
regarding your LTA: I haven’t heard the Z40. I’ve heard the microzotl drive my fathers ls3/5a speakers, and he’ll soon have AN-K’s to test, but my impression of LTA is a precise, light, airy sound quality, which I imagine would pair quite well with AN speakers. I’ve owned the AN Cobra and now own the P3 Tonmeister, but have tested Pass Sit 4’s, which may offer a similar sound to LTA, and they sound excellent with AN speakers… less warm and fleshy than AN amps, but pure sounding - clean and clear.