The index divisor free prime conjecture

Let $(d,k)$ be a tuple with $k \leq \left\lceil \frac{d-2}{2} \right\rceil$. An index divisor free prime for $(d,k)$ is a prime $p$ such that $p>k$, $p$ divides $d-r$ for some $r\leq k$, and $r$ does not divide $v_p(d-r)$, where $v_p$ is the normalized $p$-adic valuation.

Index divisor free prime conjecture. Except when $(d,k)=(27,3)$, there exists an index divisor free prime for $(d,k)$.

The paper proves existence for all but finitely many explicitly computable tuples, and this conjecture asserts that the sole exception is $(27,3)$.