Scott-set characterization of transcendence for countable recursively saturated models

Let $M$ be a countable recursively saturated model of $\mathrm{PA}$, and let $\operatorname{SSy}(M)$ denote its standard system. Transcendence characterization conjecture. There is a property of Scott sets such that

$$M \mdls \mathrm{PA}$$

is transcendent if and only if $\operatorname{SSy}(M)$ has that property. The question concerns whether transcendence for countable models can be characterized by saturation properties; the text conjectures this partial answer for models of $\mathrm{PA}$, while noting that the analogous assertion is expected to fail in other settings.