Wilf-class and palindromic-prefix enumeration conjecture for Hertzsprung patterns

Let $\mathcal{S}_k$ denote the set of Hertzsprung patterns of length $k$, and let $\Omega(\sigma,\sigma)$ be the autocorrelation polynomial of $\sigma\in\mathcal{S}_k$. For a palindrome $w\in\{0,1\}^k$, let $P(w)$ be the set of lengths of its palindromic prefixes. Define

$$a_k=|\{\Omega(\sigma,\sigma):\sigma\in\mathcal{S}_k\}|,$$

and let $b_k$ be the number of distinct sets $P(w)$ for palindromes $w\in\{0,1\}^k$. Wilf-class and palindromic-prefix enumeration conjecture. For $k\geq 3$, one has

$$a_k=b_{k+1}.$$

The equality is supported by the computed values through $k=15$, which agree with the OEIS sequence A304178 from $k\geq3$; a general proof is not supplied.