A positive proportion of locally soluble quartic Thue equations is globally insoluble

For any fixed nonzero integer $h$, we show that a positive proportion of integral binary quartic forms $F$ do locally everywhere represent $h$, but do not globally represent $h$. We order classes of integral binary quartic forms by the two generators of their ring of $\textrm{GL}_{2}(\mathbb{Z})$-invariants, classically denoted by $I$ and $J$.