Laddawn.com : Josephine text/chat conversation feature

I visited a website I heard about on NPR, Josephine.com,  which is like an Uber for purchasing home cooked meals as takeout.  (So far only available in CA.)

I noticed a feature similar to the one we're thinking of developing and thought it would be of interest to you. There are some interesting ideas here - some that I would want to experiment with, others not so much.

Homepage, first visit - answers question, "What would you see for the very first time, if you haven't had any conversations yet?" Answer: a semi-personalized invitation to engage.  The feature is anchored to the bottom right corner, so is always in view as you scroll.  I liked its unobtrusive placement; I do think the red dot with "1" in it made it noticeable to me in spite of the placement.

When I hover over the text bubble, an 'x' button appears - clicking it makes the text bubble go away ("Emily" remains).

What I see after clicking on Emily - the same text from the bubble, no longer truncated, and now within a familar text/chat/IM interface.

And so I decided to reply... And give them my email address to be notified of followup:

After clicking the minimize button, I'm brought back to the homepage, Emily is still there, but the "1" for new message is gone:

If I "X" out of the conversation instead, I just see a generic conversation icon - I've lost Emily. 

Clicking on the generic icon brings me back to the texting interface, where I see a truncated version of the prior conversation as well as an invitation to start a new conversation (not shown - clicking on the truncated conversation brings me fully back into it):

After clicking on New Conversation:

Note (above), the "hamburger" menu in the top left appears in some screens and not others, and behaves as sort of as a toggle between current conversations and new conversations (IMHO - not very intuitive and not very consistent with the behavior of this symbol - I'd expect to see a menu or be prompted make some deliberate choices or something, not to be toggled between views).

As you can see from the fine print below the response field, the feature is powered by “Intercom.” It is based on their “Aquire” tool,  which advertises itself as live chat for converting visitors to customers. They have the same feature on their homepage, but it seems configured to run a little differently. For example, instead of having a message waiting for you, they ping you (includes a sound effect) after you've had your browser open to their site for a few minutes.