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We study the task of noiseless linear regression under Gaussian covariates in the presence of additive oblivious contamination. Specifically, we are given i.i.d.\ samples from a distribution $(x, y)$ on $\mathbb R^d \times \mathbb R$ with $x \sim \mathcal N(0,I_d)$ and $y = x^\top \beta + z$, where $z$ is drawn from an unknown distribution that is independent of $x$. Moreover, $z$ satisfies $\mathbb P[z = 0] = \alpha>0$. The goal is to accurately recover the regressor $\beta$ to small $\ell_2$-error. Ignoring computational considerations, this problem is known to be solvable using $O(d/\alpha)$ samples. On the other hand, the best known polynomial-time algorithms require $\Omega(d/\alpha^2)$ samples. Here we provide formal evidence that the quadratic dependence in $1/\alpha$ is inherent for efficient algorithms. Specifically, we show that any efficient Statistical Query algorithm for this task requires VSTAT complexity at least $\tilde{\Omega}(d^{1/2}/\alpha^2)$.