Related question: [1] ring-valued points of locally ringed spaces

There is a natural functor from locally ringed spaces to presheaves on affine schemes: $$ \begin{aligned} F \colon \text{LRS} &\to \text{Psh}(\text{Aff}) \\ X &\mapsto X(\_) \end{aligned} $$

As Martin showed in [1], this functor is not full. He asks whether it is also not faithful.

I want to pose two related questions.

Q1 Is there a non-scheme locally ringed space $X$, such that $X(\_)$ is representable by a scheme? (In other words, are there an LRS $X$, and a scheme $Y$, such that $X \not\cong Y$, but $F(X) \cong F(Y)$.?)

Q2 Is the functor $F$ essentially surjective? (In other words, are all presheaves $\mathcal{F}$ in $\text{Psh}(\text{Aff})$ “representable” by locally ringed spaces?)

My gut feeling is that the answer to Q1 is “yes” and the answer to Q2 is “no”. I do not really have a good feeling where to look for an answer (but the Christmas break brought me a severe cold, which might keep me from seeing trivial (counter)examples).

Related question: [1] ring-valued points of locally ringed spaces

There is a natural functor from locally ringed spaces to presheaves on affine schemes: $$ \begin{aligned} F \colon \text{LRS} &\to \text{Psh}(\text{Aff}) \\ X &\mapsto X(\_) \end{aligned} $$

As Martin showed in [1], this functor is not full. He asks whether it is also not faithful.

I want to pose two related questions.

Q1 Is there a non-scheme locally ringed space $X$, such that $X(\_)$ is representable by a scheme? (In other words, are there an LRS $X$, and a scheme $Y$, such that $X \not\cong Y$, but $F(X) \cong F(Y)$.?)

Q2 Is the functor $F$ essentially surjective? (In other words, are all presheaves $\mathcal{F}$ in $\text{Psh}(\text{Aff})$ “representable” by locally ringed spaces?)

My gut feeling is that the answer to Q1 is “yes” and the answer to Q2 is “no”. I do not really have a good feeling where to look for an answer (but the Christmas break brought me a severe cold, which might keep me from seeing trivial (counter)examples).