The reduction that makes $1 \mid r_j \mid L_{\max}$ NP-hard (concept $Sanity$, from lax-391470 with the job lengths $\{3, 2\}$) produces only instances whose processing times are $2$ or $3$; on a word that is not such an instance it produces the instance without jobs. So the problem is NP-hard already on the instances with $p_{\max} \le 3$, and on those with at most two distinct processing times.

Theorem. $1 \mid r_j \mid L_{\max}$ is para-NP-hard parameterized by $p_{\max}$, and parameterized by the number $\#p$ of distinct processing times, on the standard reading.

The explicit reductions of lax-391470 (Cook–Levin to satisfiability, then the formula instance, then the stacked instance) produce release times and deadlines at most a constant times the number of jobs: the time line of the formula instance has length $2nS$ with $S$ linear in the number $m$ of clauses, the stacked instance has $n + (2n-1)m \ge n(m+1)$ connected pairs, five jobs each, and stacking adds at most $(p + 2q)$ per pair. So every number of the emitted word is linear in its length.

Theorem. $1 \mid r_j \mid L_{\max}$ is strongly NP-hard on the standard reading.