Converse to the parking-function criterion for strongly common radical solutions

Let $\mu=(\mu_1\geq\mu_2\geq\dots\geq\mu_r)$ be a partition. It admits a strongly common radical solution if, for some integer $m$, a common radical solution exists for any choice of $r$ distinct roots. For a permutation $\pi$ of $\mu$, write $(\pi\circ\mu)_i$ for the $i$-th entry of the resulting partition, and let $|\mu|$ denote the sum of the parts. Converse to the parking-function criterion. A necessary and sufficient condition for $\mu$ to admit a strongly common radical solution is that there exists a sequence $\{a_1,\ldots,a_r\}$ of positive integers such that the number of different permutations $\pi$ of $\mu$ satisfying $(\pi\circ\mu)_i\geq a_i$ is at least

$$|\mu|-\sum_{i=1}^r a_i+2.$$

This is presented as the converse to the sufficient condition in the preceding theorem; the source describes the claim as strongly suspected and discusses it among open problems, with only special cases currently handled.