Strictly balanced factors are as informative as unbalanced factors

Let $\Sigma=\{\mathtt{a},\mathtt{b}\}$, and let $\Sigma_{sb}^{n}$ denote the strictly balanced words of length $n$. For a word $w$, write $\operatorname{ScatFact}_k(w)$ for its set of length-$k$ scattered factors. Two words are reconstructed from their $k$-spectra when equality of the relevant scattered-factor sets forces the words to be equal. Strictly balanced reconstruction conjecture. For strictly balanced words $w$, the strictly balanced scattered factors are just as informative as the unbalanced scattered factors: for every $k\in\mathbb{N}$, if $k'=k+1$ for odd $k$ and $k'=k+2$ for even $k$, and $w,w'\in\Sigma_{sb}^{2k}$ satisfy

$$\operatorname{ScatFact}_{k'}(w)\cap\Sigma_{sb}^{k'}=\operatorname{ScatFact}_{k'}(w')\cap\Sigma_{sb}^{k'},$$

then $w=w'$. This conjecture concerns reconstruction of strictly balanced binary words from their strictly balanced scattered factors; the source states that it is unresolved, while proving it for words with at most two blocks of $[?]$-letters and, by symmetry, at most two blocks of the other letter.