The New Doomsday Conjecture for $Sq^0$-families

Let $A_*$ denote the dual Steenrod algebra, and let $Sq^0$ act on $\operatorname{Ext}_{A_*}(\mathbb{F}_2,\mathbb{F}_2)$. An $Sq^0$-family is a sequence

$$\{x,Sq^0(x),\ldots,(Sq^0)^n(x),\ldots\}.$$

New Doomsday Conjecture. For any $Sq^0$-family in $\operatorname{Ext}_{A_*}(\mathbb{F}_2,\mathbb{F}_2)$, only finitely many elements survive to the $E_\infty$-page of the classical Adams spectral sequence. The conjecture is presented as a modification of the disproved Doomsday Conjecture; its resolution is not specified in the source.