Didn't seem like it saved my state. I have browser set to delete cookies on close. I entered an exception for elgrandiablo.itch.io, but only after two "runs". Firefox now says it has 7 cookies, but I still start from the beginning. (I'd forgotten where to collect all the souls - took me a while.)
Seems like I can get just the tiniest bit farther after the second gate. My avatar leaves the screen but I don't get a new map. At least once I was able to get the avatar to come back to the screen with cursor left. If I let go of the keys as soon as I leave the screen, apparently the Snowrisu kills me and I get the try-again message. Try again works OK - that is to say I can evade the snowrisu again and open the 2nd gate again. If I hold down cursor right after my avatar leaves the screen, apparently it gets to shelter(?). The snowrisu still shakes the screen but I don't get the message that I've failed and try again.
Dunno about the no input allowed messages. On the first run I thought I was getting those when I hit cursor up in a disallowed spot. The second run I didn't see any until the second gate.
I might try again from Windows on Tuesday.