This is completely out of curiosity.
I wonder if there has been any recent progress reported on the Finiteness or New Doomsday conjecture, in the form of a talk, preprint or possibly a paper?
Just to recall that the conjecture says that any $Sq^0$-family $$\{x,Sq^0(x),Sq^0Sq^0(x),...,(Sq^0)^n(x),...\}$$ in the Adams spectral sequence $\mathrm{Ext}_A(\mathbb{F}_2,\mathbb{F}_2)$ converging to ${_2\pi_*^s}$ detects only finitely many elements. Here $A$ is the mod $2$ Steenrod algebra.
I also wonder if the conjecture has any equivalent formulations other than the one proposed by Minami? and if there is any written work on the relation between this conjecture and other important problems related to the Steenrod algebra (I am interested in the case of the prime $p=2$)?
I will appreciate any advise on this.
EDIT: I have recently looked at Bruner's Edinburgh talk, also cited in the below answer, and therein he mentions relations to some other problems - but the details of these relations I have not seen. Also, Minami in his paper in which he formulates the New Doomsday Conjecture, he mentions Wood's work on the Hit problem for $B\mathbb{F}_2^{\times n}$ as a new ingredient in his proof. In this case, what I like to know is that if the hit problem and the Finiteness Conjecture are related in a systematic way and whether or not if the details of this relation exists somewhere in the literature?!
