Let $k\geq2$. Is it true that, for any distinct integers $1 < n_1 < \cdots < n_k$ such that $\sum_{i=1}^k \frac{1}{n_i} = 1$, we must have $\max(n_{i+1} - n_i) \geq 3$?

The conjecture erdos_287 would follow for all but at most finitely many exceptions if it were known that, for all large $N$, there exists a prime $p \in [N, 2N]$ such that $\frac{p+1}{2}$ is also prime.

More precisely: if the prime conjecture holds, then there exists $k_0$ such that for all $k \geq k_0$, any Egyptian fraction representation of $1$ with $k$ terms and all terms $> 1$ must have $\max(n_{i+1} - n_i) \geq 3$.