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Let $A=KQ/I$ be a quiver algebra with relations in $I$ having only coefficients 1 or -1. This implies that $A=FQ/I$ is defined over any other field $F$ (possibly of even another characteristic).

Question: Is $A$ derived equivalent to a hereditary algebra $KQ$ over a field $K$ if and only if $A$ is derived equivalent to $FQ$ over another field $F$?

This is true in case $Q$ is of Dynkin type, but even here I know no elementary reason (is there a nice argument in this case?).

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