Yes, this is true. If the rental property is located in South Carolina you will report the income from that piece of real estate in a SC 1065 return. Keep in mind though that the income may populate into your DC 1065 from the federal return also and you may need to exclude this from DC income so you won't get taxed in both states.
If there is a income loss from that piece of real estate, you have an option whether to file it or not, unless there are estimated taxes as you mention. This is a consideration since there is no other SC income to report on the state return.
Not knowing SC laws, it may be wise to file just to escape scrutiny from the SC Dept of Revenue, even though it is an income loss. Some jurisdictions are stranger than others.
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