When extended query protocol ends?

While taking care of a Pgpool-II trouble report from user [1], I found
an interesting usage pattern of the extended query protocol. In my
understanding a series of queries in the extended query protocol is
ended by a sync message. Then one ReadyForQuery response comes for one
sync message. However this does not apply if simple query message is
sent instead of sync.

Below is outputs from "pgproto" command coming with Pgpool-II.
(Lines starting "FE" represents a message from frontend to backend.
Lines starting "BE" represents a message from backend to frontend.)

FE=> Parse(stmt="", query="SET extra_float_digits = 3")
FE=> Bind(stmt="", portal="")
FE=> Execute(portal="")
FE=> Parse(stmt="", query="SET extra_float_digits = 3")
FE=> Bind(stmt="", portal="")
FE=> Execute(portal="")
FE=> Query (query="SET extra_float_digits = 3")
<= BE ParseComplete
<= BE BindComplete
<= BE CommandComplete(SET)
<= BE ParseComplete
<= BE BindComplete
<= BE CommandComplete(SET)
<= BE CommandComplete(SET)
<= BE ReadyForQuery(I)
FE=> Terminate

As you can see, two "SET extra_float_digits = 3" is sent in the
extended query protocol, then one "SET extra_float_digits = 3" follows
in the simple query protocol. No sync message is sent. However, I get
ReadyForQuery at the end. It seems the extended query protocol is
ended by a simple query protocol message instead of a sync message.

My question is, is this legal in terms of fronted/backend protocol?
If it's legal, I think we'd better to explicitly mention in our
document. Otherwise, users may be confused. For example, in
"55.2.4. Pipelining":

"When using this method, completion of the pipeline must be determined
by counting ReadyForQuery messages and waiting for that to reach the
number of Syncs sent."

Apparently this does not apply to the above example because there's 0
sync message.

Best reagards,

[1] https://www.pgpool.net/pipermail/pgpool-general/2023-December/009051.html
--
Tatsuo Ishii
SRA OSS LLC
English: http://www.sraoss.co.jp/index_en/
Japanese:http://www.sraoss.co.jp