Is it true that for all $m\geq n+k$ \[M(n,k) \neq M(m,k)?\]

OPEN

Let $M(n,k)=[n+1,\ldots,n+k]$ be the least common multiple of $\{n+1,\ldots,n+k\}$.

Is it true that for all $m\geq n+k$ \[M(n,k) \neq M(m,k)?\]

In general, how many solutions does $M(n,k)=M(m,l)$ have when $m\geq n+k$ and $l>1$? Erdős expects very few (and none when $l\geq k$).

The only solutions Erdős knew were $M(4,3)=M(13,2)$ and $M(3,4)=M(19,2)$.

In [Er79d] Erdős conjectures the stronger fact that (aside from a finite number of exceptions) if $k>2$ and $m\geq n+k$ then $\prod_{i\leq k}(n+i)$ and $\prod_{i\leq k}(m+i)$ cannot have the same set of prime factors.