Possible similarities between a poset and its poset of endomorphisms

Question

The category variant of this question is posted separately here.

For any poset $X$, write $E(X)$ for the poset of endomorphisms of $X$ (with $f\le g$ for $f,g\in E(X)$ if and only if $f(x)\le g(x)$ for all $x\in X$), and consider the following properties a poset $X$ may or may not have:

(P1) $X$ is a singleton,

(P2) $E(X)$ is isomorphic to $X$,

(P3) there is an injective morphism $E(X)\to X$,

(P4) there is a surjective morphism $X\to E(X)$.

Clearly (P1) implies (P2), and (P2) implies (P3) and (P4): $$ \begin{matrix} &&1\\ &&\downarrow\\ &&2\\ &\swarrow&&\searrow\\ 3&&&&4. \end{matrix} $$

Denote by (Qij) the question "Does (Pi) imply (Pj)?".

student9909 asked Question (Q21) here. (student9909 accepted an answer which doesn't answer the question. I find this very confusing. As far as I know, the question is still open.) Let us ask also:

Question (Q31) Does (P3) imply (P1)?

Question (Q41) Does (P4) imply (P1)?

Question (Q32) Does (P3) imply (P2)?

Question (Q42) Does (P4) imply (P2)?

Question (Q34) Does (P3) imply (P4)?

Question (Q43) Does (P4) imply (P3)?

Answer 1

In fact the four properties are equivalent. This follows immediately from Theorem 3 in

Roy O. Davies, Allan Hayes and George Rousseau, Complete lattices and the generalized Cantor theorem, Proc. Amer. Math. Soc. 27 (1971), 253–258.

This reference was pointed out to me by user bof in a (now deleted) comment. More precisely bof gave a link to this answer of them.