When an Internet router routes a unicast IP frame from one Ethernet
LAN to another Ethernet LAN, the router replaces the frame's
Destination MAC address with the MAC address of the next router (or of
the desination host, if this is the last hop), replaces the frame's
Source MAC address with this router's own MAC address, and decrements
the TTL.

When a router is routing an IP Multicast frame, the destination MAC
address does not change, because the MAC address of an IP Multicast
frame is computed in a deterministic way from the IP address.

My question is this: When a router is routing an IP Multicast frame,
does it still replace the original source MAC address with its own MAC
address when forwarding the frame?  My guess is that it does, but I
have been unable to find any document which confirms my own guess.

I'd like a reference to an official standard, such as an RFC, which
clearly states, yes or no, whether the router puts its own MAC address
in the source MAC field when routing multicast, just as it does in
unicast.  If there is no official standard that specifies this
behavior (although I would assume there must be), then a reference to
a white paper from a well-known company like Cisco would be almost as
good.