Why would the service close? It's supported by LinkedIn and AFAIK they're not in the habit of shutting things down. This almost feels like a no-brainer for them, especially given the move to mobile devices and locked-down apps.
Edit: Have I missed the point? I'm sure LinkedIn is a little more cautious about such changes than your average newly-founded startup. This product gives them access to people emails which they can probably glean a lot of info from. They already try to get access into email accounts (username + password) via the LinkedIn webapp so in a way this is just extending that to mobile. The only reason I can imagine them shutting it down is if no-one uses it (in which case, no one will complain).
Think of it like a value proposition. Is the (dubious IMO) convenience of having Linkedin profiles in your email worth the cost of Linkedin having the content of your email? Even if they pinky-swear to never read it, don't forget that this proxy email server would be, overnight, one of the most valuable corporate espionage targets in the world.
(If yes, you should probably ditch reading email and do something more productive with your time, like picking up cans.)
Fair enough but as I tried (and obviously failed) to caveat, these are not shutdowns I really knew about. I clearly wasn't a user of those products so I didn't feel any pain when they went away.
Perhaps my claim is weak and my knowledge limited but I find it more ridiculous to claim the inevitability of a closure before a product is even being used.
NB I feel I should add that I'm not going to be a user of this product as I won't hand over email access. However, I can imagine many people who don't think the same finding it very useful.