Pointfree funcoids between join-semilattices conjecture

Today I’ve come up with the following easy to prove theorem (exercise!) for readers of my book:

Theorem If there exists at least one pointfree funcoid from a poset to a poset then either both posets have least element or none of them.

This provokes me to the following conjecture also:

Conjecture If there exists at least one pointfree funcoid from a poset to a poset , then either both or none of these two posets are join-semilattices.

If the conjecture comes up true, it would allow some simplification of some theorem conditions in my book, as there would no more necessity to claim that both source and destination are join-semilattices as I use in some of my theorems.

I do not expect that this conjecture will be particularly difficult, I have not yet invested my time into solving it.