It has been known for many years that soft radiation can give
potentially large double-logarithm corrections to pt-broadening
of a high-energy particle traveling through QCD matter, but that this
soft radiation correction can be absorbed into an effective value
$\hat q_{\rm eff}$ for the medium pt-broadening parameter
$\hat q$ . Here "soft'' means high energy compared to medium scales but
soft compared to the original high-energy particle traveling through
the medium. A similar situation arises in the case of soft
corrections to hard splitting of a high-energy particle, such as hard
$g \to gg$, where double logarithms can also be absorbed using the
same effective $\hat q_{\rm eff}$. In this talk, I discuss whether
this universality continues to hold at the level of potential large,
subleading single-logarithm corrections. (Spoiler: It does,
but the correspondence is interestingly indirect.)