Minkowski sums with convex curves without pointwise Fourier decay

Let $Γ\subset\mathbb R^2$ be a compact convex graph and define \[ T(Γ) = \inf \left\{ t: \dim_{\mathrm H}(E)>t \Longrightarrow |E+Γ|>0 \text{ for every compact }E\subset\mathbb R^2 \right\}. \] For a graph over an interval of positive length the smallest possible value is $T(Γ)=1$. We ask whether th