"Filial Correction"?   Nope... just plain old fashioned self-righteous hubris.  The signers would better spend their time prayerfully examining their own conscience and motives,  and repenting of the mortal sin of calumny.    Could it be that Pope Francis,  like Jesus,  touched a nerve with the rule-bound religionists...  so now they attack him to assuage their own guilt.   You and they protest too much.

 To some who were confident of their own righteousness and looked down on everyone else, Jesus told this parable.   Luke 18:9-14.