Realizability over Q((x)) of the form q′=(1,1,1,5,x,−5x) as a genus-2 Witt class

Determine whether the quadratic form q′=(1,1,1,5,x,−5x) in I2(Q((x))) is realizable as the Witt class of a flat SL(2, Q((x)))-bundle over a closed oriented surface of genus 2.

Background

After showing that a different 6-dimensional form q=(1,1,1,7,x,−7x) over Q((x)) cannot occur as the Witt class over a genus-2 surface (Proposition 13.1), the authors consider the variant q′=(1,1,1,5,x,−5x).

Their previous non-realizability argument relies on anisotropy of 2q; however, for q′ they note that 2q′ is isotropic, so that argument does not apply, leaving the realizability question unresolved.

The question targets the existence of a genus-2 surface representation π1(E2)→SL(2,Q((x))) whose Witt class precisely equals the class of q′ in I2(Q((x))).