See post below. It does via a persistent cookie but maybe you have two tabs open and then the second one will negate the first and you shouldn't have two tabs open anyway, it isn't supported and never has been.
Just using the one tab i use 2 panes, how do you set a persistent cookie on Chrome ?
Settings > Advanced > Content Settings > and make sure for Cookies that the first radio button is set, not the others. The last checkbox is up to you but I always check that ....
However, I think the problem probably lies somewhere else ... I just ran it on Chrome here (I normally use Waterfox as it's more stable for me) and closing/restarting it doesn't lose the Show Status setting...