Generalized shifted-sequence construction for regular Stanley sequences

Let $S(A)=\{a_n\}$ be a regular Stanley sequence with core $\{a'_n\}$, shift index $\sigma$, and character $\lambda$. Let $k$ be adequate and suppose that

$$a_{2^{k-1}-\sigma+i}=a_{2^{k-1}-\sigma}+a'_i$$

for all $0\le i<2^{k-1}$. Let $\ell$ be the minimal adequate integer for the core, and let $c$ satisfy

$$\lambda\le c\le a_{2^k-2^\ell-\sigma}+a'_{2^k}-a_{2^k-\sigma}-\lambda.$$

Generalized shift conjecture. Then $S_k(c,A)$ is defined and is a regular Stanley sequence with core $S(A)$. This would strengthen the preceding construction theorem for shifted Stanley sequences; the source presents it as an unproved stronger statement.