Trace counting is #P-complete at independence-cover number one

Exact statement

With the slice length encoded in unary, exact counting of the Mazurkiewicz traces meeting a length slice of a DFA language is $\#\mathrm P$-complete even for the fixed independence graph with the single edge $a\mathrel{\mathbb I}b$. Thus the graph has vertex-cover number one, width two, maximum degree one, and matching number one. The hardness persists although every accepted reduction word has exactly one occurrence of the chosen cover letter. At cover number zero, traces are words and DFA length-slice counting is polynomial.