Corollary: Support away from the apex

In the open set $t,s>0,R>0$, the support of $\mathrm{Im}K$ is exactly

The direct cone carries the displayed delta contribution. The reflected cone is a jump locus of the locally integrable contribution, not an additional delta distribution.

Proof. The formula vanishes in the complementary open regions. It is a strictly positive smooth function on $|t-s|<R<t+s$. The closure of that region within $t,s>0,R>0$ is the displayed set, so every point of that set lies in the support. The formula itself contains no reflected delta term.