diff --git a/Makefile.coq.local b/Makefile.coq.local
index 1a826decadc993f7c9ba8322f9922cb9b39002be..b69501acde6c39451c84ff4869b837c4e9e9e69b 100644
--- a/Makefile.coq.local
+++ b/Makefile.coq.local
@@ -24,8 +24,10 @@ test: $(TESTFILES:.v=.vo)
 .PHONY: test
 
 COQ_TEST=$(COQTOP) $(COQDEBUG) -batch -test-mode
+# These versions of Coq are known to have different output so we don't test them.
 # Need to make this a lazy variable (`=` instead of `:=`) since COQ_VERSION is only set later.
-COQ_MINOR_VERSION=$(shell echo "$(COQ_VERSION)" | grep -E '^[0-9]+\.[0-9]+\b' -o)
+# Make sure to recognize both 8.$NUM.0 and 8.$NUM+alpha.
+COQ_NOREF=$(shell echo "$(COQ_VERSION)" | grep -E "^8\.(16|17|18|20)[.+]" -q && echo 1)
 
 tests/.coqdeps.d: $(TESTFILES)
 	$(SHOW)'COQDEP TESTFILES'
@@ -41,17 +43,15 @@ tests/.coqdeps.d: $(TESTFILES)
 # - Either compare the result with the reference file, or move it over the reference file.
 # - Cleanup, and mark as done for make.
 $(TESTFILES:.v=.vo): %.vo: %.v $(if $(MAKE_REF),,%.ref) $(NORMALIZER)
-	$(HIDE)if test -f $*".$(COQ_MINOR_VERSION).ref"; then \
-	    REF=$*".$(COQ_MINOR_VERSION).ref"; \
-	  else \
-	    REF=$*".ref"; \
-	  fi && \
-	  echo "COQTEST$(if $(MAKE_REF), [make ref],) $< (ref: $$REF)" && \
+	$(HIDE)REF=$*".ref" && \
+	  echo "COQTEST$(if $(COQ_NOREF), [ref diff ignored],$(if $(MAKE_REF), [make ref],)) $< (ref: $$REF)" && \
 	  TMPFILE="$$(mktemp)" && \
 	  unset OCAMLRUNPARAM && \
 	  $(TIMER) $(COQ_TEST) $(COQFLAGS) $(COQLIBS) -load-vernac-source $< > "$$TMPFILE" && \
-	  sed -f $(NORMALIZER) "$$TMPFILE" > "$$TMPFILE".new && \
+	  sed -E -f $(NORMALIZER) "$$TMPFILE" > "$$TMPFILE".new && \
 	  mv "$$TMPFILE".new "$$TMPFILE" && \
-	  $(if $(MAKE_REF),mv "$$TMPFILE" "$$REF",diff --strip-trailing-cr -u "$$REF" "$$TMPFILE") && \
+	  $(if $(COQ_NOREF), (diff --strip-trailing-cr -u "$$REF" "$$TMPFILE" || true) , \
+	    $(if $(MAKE_REF),mv "$$TMPFILE" "$$REF",diff --strip-trailing-cr -u "$$REF" "$$TMPFILE") \
+	  ) && \
 	  rm -f "$$TMPFILE" && \
 	  touch $@
diff --git a/test-normalizer.sed b/test-normalizer.sed
index 3f9662bba47b73b0deac50eb948f084345fc0404..e26d3a5e4072ff03e7a19946c1efb08a8ffb4215 100644
--- a/test-normalizer.sed
+++ b/test-normalizer.sed
@@ -1,8 +1,2 @@
-# adjust for https://github.com/coq/coq/pull/13656
-s/subgoal/goal/g
-# merge with subsequent line for https://github.com/coq/coq/pull/14999
-/[0-9]* focused goals\?$/{N;s/\n */ /;}
 # locations in Fail added in https://github.com/coq/coq/pull/15174
 /^File/d
-# extra space removed in https://github.com/coq/coq/pull/16130
-s/= $/=/