Conjectured classification of Erdős-deep triples of arithmetic progressions

Let $\mathbb{Z}_n$ be the cyclic group of integers modulo $n$. An Erdős-deep family is a family whose distance multiset has multiplicities precisely $1,2,\dots,k-1$ for some integer $k$. Let three arithmetic progressions in $\mathbb{Z}_n$ have lengths $k_1\geq k_2\geq k_3$. Classification conjecture. An Erdős-deep family of three arithmetic progressions of these lengths exists if and only if either

$$(k_1,k_2,k_3)\in\{(4,4,3),(6,3,3)\},$$

with each of these two length triples occurring for infinitely many values of $n$, or

$$(k_1,k_2,k_3)\in\{(6,5,3),(6,6,4),(6,6,6),(7,7,3),(9,4,3),(8,7,4),(8,8,5),(10,6,4),(12,4,4),(13,5,3),(13,7,4),(16,5,4),(21,6,4)\},$$

with each of these length triples occurring for a finite number of values of $n$. The conjecture would complete the classification of Erdős-deep triples of arithmetic progressions; the paper reports that it is based on a computer search of small parameters, while the cases of two progressions have already been classified.