Is the following sentence equivalent to $\sf AC$ over the rest of axioms of $\sf ZF$?

For each infinite set $X$: if for all $y \in X$ we have $|y| < |X|$, then $| \bigcup X|\leq |X| $?

Note: The cardinality function $||$ here is defined after Scott's.

If not, then to which of the known choice principles this is equivalent to?

Is the following sentence equivalent to $\sf AC$ over the rest of axioms of $\sf ZF$?

For each infinite set $X$: if for all $y \in X$ we have $|y| < |X|$, then $| \bigcup X|\leq |X| $?

Note: The cardinality function $||$ here is defined after Scott's.

If not, then to which of the known choice principles this is equivalent to?

This comes in connection to this question.

Is the following sentence equivalent to $\sf AC$ over the rest of axioms of $\sf ZF$?

For each infinite set $X$: if for all $y \in X$ we have $|y| < |X|$, then $|X \cup \bigcup X|=|X| $?

Note: The cardinality function $||$ here is defined after Scott's.

If not, then to which of the known choice principles this is equivalent to?

Is the following sentence equivalent to $\sf AC$ over the rest of axioms of $\sf ZF$?

For each infinite set $X$: if for all $y \in X$ we have $|y| < |X|$, then $| \bigcup X|\leq |X| $?

Note: The cardinality function $||$ here is defined after Scott's.

If not, then to which of the known choice principles this is equivalent to?