Circular chromatic index of small graphs

We systematically determine circular chromatic index of small graphs and multigraphs with maximum degree $4$, $5$, $6$ (and also their number for a given small order). We construct several infinite families of such graphs with circular chromatic index in the set $\{\mathrm{\Delta }+1\mathrm{/}2,\mathrm{\Delta }+2\mathrm{/}3,\mathrm{\Delta }+3\mathrm{/}4$, $\mathrm{\Delta }+1\}$. Our results refute edge-connectivity variants of the ``Upper Gap Conjecture’’ (about the non-existence of graphs with circular chromatic index just below $\mathrm{\Delta }+1$).