Vim shortcuts not working inside tmux

I recently discovered Sakura and I'm quite liking it overall. But shortcuts for switching splits in Vim inside tmux aren't working. Specifically, C-w following by a direction [hjkl] to switch between splits. It seems that vim doesn't receive the C-w since when I press the direction, the cursor just moves.

I'm running Sakura 3.3.4 on Ubuntu MATE 16.04. This works fine with vim running under Sakura outside tmux. Shortcuts work fine in Vim inside tmux using mate-terminal.

Any help sorting this out would be appreciated :)