Has it been conjectured that all $k$-multiperfect numbers are multiples of $k$?

A quick glance at the list of the first $ k $ -multiperfect numbers for small $ k $ makes me think that all $ k $ -multiperfect numbers are multiples of $ k $ , which is a generalization of the famous "no odd perfect number" conjecture. Has this conjecture appeared so far ? If yes, what are the results towards it so far ?

I am writing this answer so that this question does not remain in the unanswered queue.

In a comment to my answer to a closely related question, Peter asserted that $459818240$ is $3$-perfect, but not divisible by $6$.

WolframAlpha computational verification that $459818240$ is $3$-perfect

WolframAlpha computational verification that $459818240$ is indivisible by $6$

This settles the open question as to whether all $k$-perfect numbers are divisible by $k!$.

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