Example of #P-intermediate problem The previous question

Do there exist intermediate problems (in the sense of Ladner's Theorem) for FP vs. #P? I assume that something is known, because I read some papers concerned with FP/#P dichotomies. However, I couldn't find a reference.

An Answer to the previous question

Use Schöning's theorem:


  Let $A_1$, $A_2$ be recursive sets and $C_1$, $C_2$ be classes of recursive sets with the following properties:
  
  
  $A_1 \notin C_1$, $A_2 \notin C_2$
  $C_1$ and $C_2$ are recursively presentable, 
  $C_1$ and $C_2$ are closed under finite variations. 
  
  
  Then there exists a recursive set $A$ such that:
  
  
  $A \notin C_1$, $A \notin C_2$, 
  if $A_1 \in \mathsf{P}$ and $A_2\notin \{ \emptyset, \Sigma^* \}$, then $A \leq^{\mathsf{P}}_m A_2$. 
  


For the purposes of counting dichotomy theorems, the two relevant classes of decision problems are $\text{P}$ and $\text{P}^{\#\text{P}}$.

My question


  Is there a concrete example of #P intermediate problem under some plausible assumption?
  More specifically, is there an explicit function $F$ satisfying the following conditions?
  $F\notin \mathsf{FP}$ and $F$ is not  $\# \mathsf{P}$-complete.
Assuming $\mathsf{PH}$ does not collapse and that Graph Isomorphism is not in $\mathsf{P}$, then $\# GI$ (the counting version of graph isomorphism) satisfies your conditions. This is because $\# GI \equiv_m^p GI$.