Finiteness conjecture for equal radicals of consecutive products

Let $k$ and $\ell$ be integers greater than $1$, and let $m$ and $n$ be integers. Define

$$R_{k}(m)=\operatorname{rad}\bigl(m(m+1)\cdots(m+k-1)\bigr),$$

where $\operatorname{rad}(r)$ is the largest squarefree divisor of $r$. Finiteness conjecture. If $(k,\ell)\neq(2,2)$, then the equation

$$\operatorname{rad}\bigl(m(m+1)\cdots(m+k-1)\bigr)=\operatorname{rad}\bigl(n(n+1)\cdots(n+\ell-1)\bigr)$$

has only finitely many solutions. The conjecture concerns the expected rarity of pairs of consecutive products with equal radicals; the exceptional case $(k,\ell)=(2,2)$ is explicitly excluded, and the source gives no resolution.