The arithmetic-progression partition heaviness conjecture

For positive integers $i,s,k$, let $\llbracket i+(k-1)s,\ldots,i+s,i\rrbracket$ denote the partition whose parts form a decreasing arithmetic progression with initial part $i+(k-1)s$, common difference $s$, and final part $i$. A partition is heavy if it has the heaviness property defined for partitions in the paper. Arithmetic-progression partition heaviness conjecture. If $i$, $s$, and $k$ are positive integers, then

$$\llbracket i+(k-1)s,\ldots,i+s,i\rrbracket$$

is heavy. This proposes a further infinite family of heavy partitions, motivated by the patterns observed in the paper; the source gives no proof or resolution.