Nonexistence conjecture for prime $v$-palindromes

Let $v\colon\mathbb{N}\to\mathbb{Z}$ be the additive function defined by $v(p)=p$ for primes $p$ and $v(p^\alpha)=p+\alpha$ for prime powers $p^\alpha$ with $\alpha\geq2$. For integers $n\geq1$ and $b\geq2$, call $n$ a $v$-palindrome in base $b$ if $b\nmid n$, $n\neq r_b(n)$, and $v(n)=v(r_b(n))$, where $r_b(n)$ is the integer obtained by reversing the base-$b$ digits of $n$; in base $10$, call it a $v$-palindrome. Nonexistence conjecture for prime $v$-palindromes. There are no prime $v$-palindromes. The claim is motivated by extensive computer calculations and complements known infinite families of composite $v$-palindromes; whether any prime examples exist remains open.