Part 4 is a multipart, not an html part.
notmuch show --format=raw --part=3 'id:87liy5ap00.fsf@yoom.home.cworth.org' >OUTPUT
test_expect_equal_file OUTPUT embedded_message
-test_begin_subtest "--format=raw --part=4, rfc822's html part"
+test_begin_subtest "--format=raw --part=4, rfc822's multipart"
notmuch show --format=raw --part=4 'id:87liy5ap00.fsf@yoom.home.cworth.org' >OUTPUT
cat <<EOF >EXPECTED
<p>This is an embedded message, with a multipart/alternative part.</p>