Just went through this process myself. It's only one point of data, but might be useful to you.
I found (sorry in advance to for the length of the post)...
1) The nic wasn't an issue.
I just cut down by halves month on month (12mg to 6 to 3 to 1). Stayed on 1mg mostly for the piece of mind (ie comfort of knowing I was getting nic) for a while (a few months?), but it turned out that was not really needed. With each concentration adjustment I would get a little light headed twice a day after lunch and dinner for about 4-5 days. This effect resolved on it's own in about 10 minutes. Within a week, I'd adjusted to the lower nic level... including the ultimate adjustment to zero.
One thing I did was pay attention to was
juice volumes consumed. As has been pointed out above, you're gaming yourself if you half the nic and then double the volume vaped. I thought I would have to ration myself, but I just kept my same daily routines and the volume consumed kept constant.
I DIY
juice so I increase the amount of koolada I put in a batch as the nic got weaker to compensate for throat hit. You DIY and use menthol, so same same - just up it a bit if/when your TH feels unfulfilling.
Once down to zero nic, I hung out there for a month and then switched to flavourless. I found this to be thoroughly bleh vape and after about a week just stopped. It had become uninteresting to me.
2) The physical habit gets dulled.
I bought a fidget spinner, some stress balls, and a hand exerciser (one of those things you squeeze to get Arnold hands). I was ready, by god.
Time spent using those things? Hmmm... about 20 minutes total. I found the best thing to do was just get up and walk for 5 minutes whenever I got a physical "nag". Just get up and walk around - anywhere, even in circles. Within 2 days the nags were half what they were at the outset and all but a slight murmur within 8 days or so. This was about 3 months ago now. High trigger contexts (eg drinks with friends who vape) don't get to me at all. Am I on the other side of it all? Who can say, but I do know that I have a mountain of vape gear and juice in the room with me as I write this and I haven't the slightest urge to use any of it.
tl;dr - it can definitely work, but it's part of a regime you'll prolly need to adopt