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For all integers $g \geq 6$ we prove the existence of a metric graph $G$ with$w^1_4=1$ such that $G$ has Clifford index 2 and there is no tropicalmodification $G'$ of $G$ such that there exists a finite harmonic morphism ofdegree 2 from $G'$ to a metric graph of genus 1. Those examples show thatdimension theorems on the space classifying special linear systems for curvesdo not all of them have immediate translation to the theory of divisors onmetric graphs.

A metric graph satisfying $w^1_4=1$ that cannot be lifted to a curve satisfying $\dim (W^1_4)=1$