The $3k-4$ conjecture for subsets of $\mathbb F_p$

Let $p$ be a large prime number and let $A e\varnothing$ be a \subset of $\mathbb F_p$. An additive set $A$ has the covering property if it can be covered by an arithmetic progression of length $|2A|-|A|+1$.

$3k-4$ conjecture. If

$$|A+A|\le \min\{3|A|-4,p-1\},$$

then $A$ has the covering property.

The conjecture is the analogue over $\mathbb F_p$ of Freiman's $3k-4$ theorem for finite subsets of the integers. It remains open, despite progress under additional assumptions on the extension ratio or density.