Shallow minimal reduction type conjecture

Let $W$ be the Weyl group and $\underline W$ its set of conjugacy classes. Let $\underline{\mathcal N}$ be the set of nilpotent orbits, and let

$$\operatorname{RT}_{\min}:\underline W\to\underline{\mathcal N}$$

be the map defined from generic minimal reduction types. For $[w]\in\underline W$, let $(L^{\heartsuit}\mathfrak g)^{\mathrm{sh}}_{[w]}$ denote the shallow stratum of elements of type $[w]$.

Shallow minimal reduction type conjecture. For every $[w]\in\underline W$,

$$\operatorname{RT}_{\min}(\mathfrak g)=\{\operatorname{RT}_{\min}([w])\}$$

for all $\mathfrak g\in(L^{\heartsuit}\mathfrak g)^{\mathrm{sh}}_{[w]}$. This is stated as a weaker version of the conjecture that all minimal reduction-type sets are singletons; its status is not resolved by the supplied material.