Scott sets and standard systems of models of PA

• A Turing ideal is a set $A \subseteq 2^\mathbb{N}$ which is closed under finite joins and Turing reducibility. I.e. for any $x, y \in A$, $x\oplus y$ is in $A$ as is any $z \leq_T x$.
• A Scott set is a Turing ideal $A$ such that for any $x \in A$ which codes an infinite binary tree, $A$ contains some element which codes an infinite path through this tree.
• Given a non-standard model $M$ of $\mathsf{PA}$, the standard system of $M$, denoted $\mathsf{SSy}(M)$, is the set of reals $x \in 2^\mathbb{N}$ such that $x$ is coded by some element of $M$, i.e. there is some non-standard $a \in M$ such that $x(n) = 1$ if and only if the $n^\text{th}$ prime number divides $a$.

• Scott [Sco62] proved that every countable Scott set is the standard system of a non-standard model of $\mathsf{PA}$.
• Knight and Nadel [KN82] proved that every size $\aleph_1$ Scott set is the standard system of a non-standard model of $\mathsf{PA}$. Hence the question has a positive answer under $\mathsf{CH}$.
• Gitman [Git08] proved that under the Proper Forcing Axiom ($\mathsf{PFA}$), Scott sets of any size which satisfy certain technical conditions (arithmetically closed and proper) are the standard system of a non-standard model of $\mathsf{PA}$.

Scott [Sco62] proved that every if $M$ is a non-standard model of $\mathsf{PA}$ then its standard system is a Scott set. This question asks whether the converse holds.