The uniform-slicing conjecture for alphabet-multiplier upcycles

Let $v$ be an upcycle constructed by the alphabet multiplier theorem from an upcycle $u$ and multiplier $k$. Let $n$ and $d$ be the parameters in that construction. Slicing $v$ into cyclic partial words of a specified common length, with the slicing allowed to start at any index, produces a collection of partial words.

Uniform-slicing conjecture. For every $0\leq i\leq n-d$, if the slicing length is $k^i|u|$, then the resulting collection is an upfamily.

This is presented as a stronger conjecture than the existence claim above. The paper reports computational verification for the upcycle $001\mathord{\diamond}110\mathord{\diamond}$ with multipliers $k=2,3,4,5$, while the general assertion remains unresolved.