This is a ``1-soft'' modification of this problem. We start with the necessary definitions.
Definition 1. A compactification $c\mathbb N$ of the discrete space $\mathbb N$ is called 1-soft if for any subset $A\subset\mathbb N$ with $\overline{A}\cap\overline{\mathbb N\setminus A}\ne\emptyset$ there exists a homeomorphism $h:c\mathbb N\to c\mathbb N$ such that $h(x)=x$ for all $x\in c\mathbb N\setminus\mathbb N$ and the set $\{x\in A:h(x)\notin A\}$ is infinite.
Definition 2. A compactification $c\mathbb N$ of the discrete space $\mathbb N$ is called 2-soft if for any disjoint sets $A,B\subset\mathbb N$ with $\bar A\cap\bar B\ne\emptyset$ there exists a homeomorphism $h:c\mathbb N\to c\mathbb N$ such that $h(x)=x$ for all $x\in c\mathbb N\setminus\mathbb N$ and the set $\{x\in A:h(x)\in B\}$ is infinite.
Before the formulation of a question, let us recall some known results.
Theorem (Parovichenko). Each compact Hausdorff space $K$ of weight $\le\omega_1$ is homeomorphic to the remainder $c\mathbb N\setminus \mathbb N$ of some compactification $c\mathbb N$ of $\mathbb N$.
Theorem (Hart). Under CH, each compact Hausdorff space $K$ of weight $\le\omega_1$ is homeomorphic to the remainder of a 2-soft compactification of $\mathbb N$.
Example (Dow). Under (NT) the compact space $K=\omega_1+1+\omega_1^*$ is not homeomorphic to the remainder of a 2-soft compactification of $\mathbb N$.
Question 1. Is the compact space $K=\omega_1+1+\omega_1^*$ homeomorphic to the remainder of a 1-soft compactification of $\mathbb N$?
Question 2. Is each compact Hausdorff space of weight $\le\omega_1$ homeomorphic to the remainder of a 1-soft compactification of $\mathbb N$?
Added in Edit. The answer to Question 1 is affirmative. So, only Question 2 remains open.
