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

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 tropical modification $G'$ of $G$ such that there exists a finite harmonic morphism of degree 2 from $G'$ to a metric graph of genus 1. Those examples show that dimension theorems on the space classifying special linear systems for curves do not all of them have immediate translation to the theory of divisors on metric graphs.

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