I don't think leaving it open would affect it... but maybe? The ingredient that makes it waterproof isn't something that I think would evaporate off.

I went and looked at the ink I use, I use Yasumoto Black Sumi Ink. It comes in a little green bottle with a black lid, it's about $4-6 a pop (Here it is on Amazon: https://www.amazon.com/Yasutomo-Liquid-Sumi-Ink-Black/dp/B000AMLR8M/ref=pd_lpo_1?pd_rd_i=B000AMLR8M&psc=1 )They also make one that is water soluble that looks the same. One way to tell is that water proof inks generally dry a little shiny because the shellac in them has a little sheen to it.

The Yasumoto ink needs a full 24 hours to become really smudge-proof. It's my favorite ink to work with for a brush or dip pen because it's really heavy and black, but it paints on just like watercolor. Personally I don't like Higgins or Speedball inks, something about the formulation makes them tacky/sticky if they dry in a big pool. Yasumoto inks are much more akin to watercolor won't become tacky if it pools on the page somewhere.