Let $X$ be a projective variety. As I've been told, there is a conjecture (by Cohen-Jones-Segal) which implies that the homotopy type of fibers  of $ev:\mathcal{M}_{0,1}(X,\beta)→X$ converge to the homotopy type of the double loop space of $X$. So what about higher genus? 

I believe that Segal original paper [The topology of spaces of rational functions](https://projecteuclid.org/download/pdf_1/euclid.acta/1485890033)
dealt with a genus $g$ Riemann surface and not neccesarily just the genus zero cases. But most of the later papers I could find seem to focus on holomorphic spheres... 

 1. What does the Cohen-Jones-Segal conjecture implies for the homotopy type of fibers $ev:\mathcal{M}_{g,1}(X,\beta)→X$ with $g \geq 1$? 
(I assume it should be something like: the inclusion into the space of differentiable maps from a genus $g$ surface and homology class $\beta$ to $X$ is a weak homotopy equivalence? ...)

 2. Am I correct in assuming that such a conjecture should imply that the only sources of $\pi_1$ in the fibers should come from universal Abel map to the relative Picard variety? 

Even more vaguely: 

 1. Does it make sense to try and use Costello's paper [Higher genus Gromov-Witten invariants as genus zero invariants of symmetric products](http://annals.math.princeton.edu/wp-content/uploads/annals-v164-n2-p05.pdf) to deduce such a statement from a corresponding genus zero Cohen-Jones-Segal for the product stack $S^{g+1}(X)$? 

Is there an example of work in this direction? 
 
 2. Are there any results coming from h-principle/oka theory (i.e., results about the inclusion into the space of differentiable maps $Hol(S_g,N) \to Map(S_g,N)$ being weak homotopy equivalence for certain Stein manifolds $N$ etc), which can be extended to the case of a projective target space?

( This is basically Part B of my previous question: https://mathoverflow.net/questions/265144/what-is-the-fundamental-group-of-kontsevichs-space-of-stable-maps following Jason's extremely helpful suggestions in the comments )