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We show that the tail probability of the rough line integral$\int_{0}^{1}\phi(X_{t})dY_{t}$, where $(X,Y)$ is a 2D fractional Brownianmotion with Hurst parameter $H\in(1/4,1/2)$ and $\phi$ is a$C_{b}^{\infty}$-function satisfying a mild non-degeneracy condition on itsderivative, cannot decay faster than a $\gamma$-Weibull tail with any exponent$\gamma>2H+1$. In particular, this produces a simple class of examples ofdifferential equations driven by fBM, whose solutions fail to have Gaussiantail even though the underlying vector fields are assumed to be of class$C_{b}^{\infty}$. This also demonstrates that the well-known upper tailestimate proved by Cass-Litterer-Lyons in 2013 is essentially sharp.

On the Lack of Gaussian Tail for Rough Line Integrals along Fractional Brownian Paths