In ${\sf ZFC}$ it can be easily proved that we cannot have infinitely descending sequences of cardinalities, that is, the following statement does not hold:

(DescSeq) There is a set $A$ a map $\alpha: \omega \to {\cal P}(A)$ such that for all $n\in \omega$ we have $\alpha(n+1) \subseteq \alpha(n)$, and there is no injective function from $\alpha(n)$ into $\alpha(n+1)$.

It seems to be unknown whether $\neg$(DescSeq) implies the Axiom of Choice.

Question. In ${\sf ZF}$, are there any implications between $\neg$(DescSeq), the Boolean Prime Ideal Theorem, and the Partition Principle?