Let $\chi'\_f(G)$ be the fractional chromatic index.

Based on limited experiments (up to 8 vertices and few larger graphs),
I suspect:

**Conjecture** For perfect graphs $\lceil \chi'\_f(G) \rceil = \chi'(G)$

**Conjecture 2 (new)** For cubic claw-free perfect graphs $\lceil \chi'\_f(G) \rceil = \chi'(G)$

**Conjecture 3 (new)** For claw-free perfect graphs $\lceil \chi'\_f(G) \rceil = \chi'(G)$

Sage's `fractional_chromatic_index()` is not efficient for me,
is there another implementation?

Counterexamples or proof (especially of (2)) are welcome.

Observe that the question is about **edge coloring**,
not for vertex coloring.