Mark Watkins reports that his computer has proved 1 e3 e6 to be a win for White, leaving only 1...b6 and 1...c5 as possible non-losing replies to 1 e3. I quote from his report, slightly expanded for presentation outside its original context.
"We started our work in late 2011. The long-term goal was to weakly solve the game [to find the result with best play by both sides], presumably by showing that 1. e3 wins for White. At that time, to the best of our knowledge there were 13 Black responses to 1. e3 that were proven to be losses [1...a6/a5/c6/d6/d5/e5/f6/f5/g6/h6/h5/Na6/Nf6]. We proved that 1. e3 Nc6 (early 2012) and 1. e3 b5 (August 2012) are both won for White. When relaying this information to Cătălin Frâncu (and others), he replied that he had recently shown 1. e3 Nh6 was a White win when testing a new laptop, and 2 weeks later we showed 1. e3 g5 is a White win. More recently (August 2014), we proved 1. e3 e6 is also a White win, leaving b6 and c5 as the remaining Black responses."
For further information, see Mark's web site http://magma.maths.usyd.edu.au/~watkins/LOSING_CHESS/.
But even when Losing Chess has been completely solved as a game, whether by proving 1 e3 to be a win for White or otherwise, the beautiful endgame positions which are my particular delight will be unaffected.