Erdős's conjecture on three consecutive powerful numbers

A positive integer is powerful if every prime divisor $p$ satisfies $p^2\mid n$. For an integer $N$, the three consecutive integers are $N$, $N+1$, and $N+2$. Erdős's conjecture. There is no integer $N$ such that all three of $N$, $N+1$, and $N+2$ are powerful.

This folklore conjecture is attributed to Erdős and was later reiterated by Mollin, Walsh, and Granville. It remains unproved, although it follows from the $abc$-conjecture.