The index divisor conjecture for products of consecutive integers

Let $k$ be a non-negative integer and let $n>2k+2$, and define

$$\Delta=n(n-1)\cdots(n-k).$$

Index divisor conjecture. There exists a prime $p>k$ which divides $\Delta$. Moreover, the index at which $p$ occurs in the product $n(n-1)\cdots(n-k)$ does not divide the power to which it occurs in $\Delta$, except when $(n,k)=(27,3)$.

This is a number-theoretic reformulation of the index divisor free prime conjecture. The paper presents it as a conjectural restatement because computing all exceptions is not currently feasible.