Conjecture on the power-series coefficients of the expected largest BP-factorization width

Let $E_k$ denote the limit of the expected width of the largest block-palindrome factorization of a length-$n$ word over an alphabet of size $k$, whose existence is established for every $k\geq 2$. Let $(a_i)_{i\geq 1}$ be the coefficient sequence in the formal expansion in powers of $k^{-1}$. Coefficient conjecture. For every integer $k\geq 2$,

$$E_k=1+\sum_{i=1}^{\infty}a_i k^{-i},$$

where the sequence $(a_n/2)_{n\geq 1}$ is OEIS sequence A274199. This conjecture identifies the empirically observed coefficients of the expected limiting width; the paper provides numerical evidence but does not establish the claimed coefficient sequence.