Impossible to tell, too many things will have changed between 1867 and 1914-22 - likely including Max himself. For all we know, a living Max could've been embittered by his defeat in Mexico and turned his back on liberalism, ironically becoming more of the sort of stalwart reactionary FJ would've approved of. Alternatively, he could've retained his old views and been a positive influence on his nephew Rudolf, who apparently was more liberal than FJ and clashed with the latter because of it. If the butterflies (including a good relationship with his like-minded uncle) get Rudolf to survive, well I do believe you've cracked open a jar full of butterflies over A-H's head. Either way, Max'd probably be right back to worrying about the Austrian Navy during his brother's long reign and would primarily be focused on keeping it in tip-top shape after Tegetthoff's death (unless the butterflies spare him too).
In any case, I don't think this is a good thread to discuss Maximilian's survival and the implications thereof further. After all, my original POD was that France wins the FPW and the ill-fated archduke will have already been dead for three years by the time that happens.
