Cardinality propagation for k-spectra of theta-palindromes

Let $\Sigma=\{\mathtt{a},\mathtt{b}\}$, let $w^R$ denote reversal, and let $\operatorname{ScatFact}_k(w)$ be the set of length-$k$ scattered factors of $w$. Consider the binary $\theta$-palindromic setting described in the source, with $w=\mathtt{a}\mathtt{b}^{k-1}\mathtt{a}^{k-1}\mathtt{b}$ and, for a word $w'$, $w=w'^R$. Theta-palindrome spectrum conjecture. The $k$-spectrum of $w=\mathtt{a}\mathtt{b}^{k-1}\mathtt{a}^{k-1}\mathtt{b}$ has $4(k-1)$ elements; moreover, if $w'=w^R$ has a $k$-spectrum of cardinality $\ell\in\mathbb{N}_{\geq12}$, then the scattered-factor set of $\mathtt{a}w\mathtt{b}$ has cardinality $2\frac{1}{4}\ell-5$. The conjecture is presented as a possible propagation law for spectrum cardinalities of $\theta$-palindromes and is intended to explain the behavior of later gaps in that restricted class; it is not resolved in the supplied text.