The covering-radius conjecture for small linear codes

Let $Q\subset \mathbb F_2^n$ be an $\mathbb F_2$-linear code of size $n^t$, where $t=\Theta(1)$. Its covering radius is the maximum, over $u\in\mathbb F_2^n$, of the minimum Hamming distance from $u$ to a codeword of $Q$. The covering-radius conjecture. The covering radius of $Q$ is at least

$$\frac n2-O(\sqrt n).$$

Equivalently, for each constant $t>0$, there exists a constant $c>0$ such that, for $n$ large enough, for each $\mathbb F_2$-linear code $Q\subset \mathbb F_2^n$ of size at most $n^t$, there exists $u\in\mathbb F_2^n$ whose distance from every codeword of $Q$ is at least

$$\frac n2-\sqrt{cn}.$$

This is presented as a stronger conjecture than the assertion that some coset has weight distribution bounded away from the binomial distribution, and the paper leaves the question open.