All unique keys have at least one optional column

WITH unique_columns AS (SELECT 
  n.nspname AS table_schema,
  c2.relname AS table_name,
  a.attname AS column_name,
  c.relname AS index_name
FROM 
  pg_catalog.pg_class AS c INNER JOIN pg_catalog.pg_index AS i ON i.indexrelid = c.oid
  INNER JOIN pg_catalog.pg_class AS c2  ON i.indrelid = c2.oid
  INNER JOIN pg_catalog.pg_namespace AS n ON c.relnamespace = n.oid
  INNER JOIN pg_catalog.pg_authid AS u ON  n.nspowner = u.oid
  INNER JOIN pg_catalog.pg_attribute AS a ON a.attrelid = c.oid    
WHERE c.relkind = 'i'
AND i.indisunique=TRUE
AND i.indisprimary=FALSE
AND a.attnum>=1
AND a.attisdropped = FALSE
AND (n.nspname = 'public'
	 OR u.rolname <> 'postgres')
AND i.indnullsnotdistinct = FALSE)

SELECT A.table_schema, A.table_name, Count(DISTINCT index_name) AS nr_of_nullable_keys
FROM unique_columns AS U INNER JOIN information_schema.columns A USING (table_schema, table_name, column_name)
INNER JOIN information_schema.tables T USING (table_schema, table_name)
INNER JOIN information_schema.schemata S ON A.table_schema=S.schema_name
WHERE T.table_type='BASE TABLE'
AND (A.table_schema = 'public'
OR S.schema_owner<>'postgres')
AND A.is_nullable='YES'
GROUP BY  A.table_schema, A.table_name
HAVING Count(DISTINCT index_name)=(SELECT Count(DISTINCT index_name) AS cnt FROM unique_columns AS u WHERE u.table_schema=A.table_schema AND u.table_name=A.table_name)
ORDER BY table_schema, table_name;