The loop is not evaluated

The schematic mixed retarded loop contains a scalar Hadamard line together with retarded metric and coordinate lines. If $\mathrm{supp}f\cap \mathrm{supp}g=\varnothing$, the endpoint coincidence $x=y$ is absent. This removes that particular diagonal, not every singular configuration.

At distinct null-related endpoints, a retarded kernel has both conormal orientations while a Hadamard kernel has one. Their naive product can therefore contain opposite covectors whose sum is zero. Failing this criterion is not by itself a universal nonexistence theorem, but it invalidates the claimed automatic multiplication.

A properly defined retarded quantum product would need its ordering combinations and extension data specified. The existence of ${\star }_H$ does not supply those data for this nonlocal retarded loop. Its value remains unresolved and is not used in admission, nor made finite merely by disjoint supports.