The Overfull conjecture in graph theory and $coNP$ I am not good at complexity, but got a possible
relation between a plausible conjecture in graph
theory and $coNP$.

Graph $G$ is Class 1 if it can be edge colored
with $\Delta(G)$ colors, otherwise it is Class 2 and
can be edge colored with $\Delta(G) + 1$ colors.

The Overfull conjecture (OC)
asserts


  A graph G with $\Delta (G) \geq n/3$ is class 2 if and only if it has an overfull subgraph $S$ such that $\displaystyle \Delta (G) = \Delta (S)$.


Assume OC and $\Delta (G) \geq n/3$.

This means that we have short certificate if $G$ is Class 1
or Class 2.

For Class 1 the certificate is $\Delta(G)$ edge coloring finding it is in $NP$.

For Class 2 the certificate is overfull subgraph $S$ such that $\displaystyle \Delta (G) = \Delta (S)$ and finding it is in $NP$.

This means there are no $coNP$-hard problems in this case.

There is a reduction from SAT to edge coloring
$3$-regular graphs. Encoding unsatisfiable CNF
to edge coloring is UNSAT and UNSAT is in $coNP$.


  Question:  Does the overfull conjecture and reduction
  from SAT to edge coloring $G$ with $\Delta (G) \geq n/3$
  imply $NP=coNP$?


According to a paper
edge coloring is NP-complete (possibly minor abuse of terminology) for
$r$-regular graphs for any fixed $r \ge 3$ and it gives reduction
of $G$ to $r$-regular $G'$.

Positive answer might help for counterexample to the
overfull conjecture.

$\Delta(G)$ is the maximum degree and $n$ is the number of
vertices.



EDIT Special cases of OC are proven.

According to Overfull Conjecture for Graphs with High Minimum Degree,
Michael Plantholt:


  we show that any (not necessarily regular) graph $G$ of even order $n$ that has sufficiently high minimum degree $\delta(G) \ge (\sqrt{7}/3) n$  has chromatic index equal to its maximum degree providing that $G$ does not contain an
  "overfull" subgraph, that is, a subgraph which trivially forces the chromatic
  index to be more than the maximum degree. This result thus verifies the
  Overfull Conjecture for graphs of even order and sufficiently high minimum
  degree.


EDIT 2
This paper p.2
might be related since it claims unless $NP=ZPP$, one can't
approximate $\chi'(L^k(G))$ where $L^k(G)$ is the $k-th$ power of
the line graph of $G$. (might be wrong on this since $\chi'$ is either
$\Delta$ or $\Delta+1$).
For the possible reduction you are mentioning, if you find such a reduction from SAT to this particular edge-coloring problem, and additionnally you assume OC, then it would indeed mean  NP $=$ co-NP. However, it would be very surprising:   it would mean that basically, up to encoding, you found a way to always provide a short certificate that a formula is not satisfiable.
This seems quite unlikely to reach with a simple construction.