Splicing conjecture for zeta-like tuples when q=2

Let $q=2$, and let $(s_1,\dotsc,s_k)$ and $(s_k,\dotsc,s_r)$ be zeta-like tuples. Their total weight is

$$\sum_{i=1}^r s_i.$$

Splicing conjecture. If this total weight is a power of $2$ or one less than a power of $2$, then $(s_1,\dotsc,s_r)$ is zeta-like, except when the two tuples being spliced are $(1,1)$ and $(1,1)$. The source reports no further failures in its limited data and says that the analogous general-$q$ situation appears more restrictive.