Rational antipalindromic quotient conjecture

A rational number $p/q$ has an antipalindromic quotient representation if there are antipalindromic numbers $A$ and $B$ such that $p/q=A/B$. Rational antipalindromic quotient conjecture. For every integer $p\geq 4$, there exists an integer $q<p$ such that $p/q=A/B$ has a solution in antipalindromes $A,B$.

The conjecture is based on computations for $p>q$ and $p\leq 1000$, following an algorithm that decides whether a given rational number has an antipalindromic quotient representation; the general assertion remains open.