Nonexistence conjecture for Buchsbaum homogeneous integral domains of minimal multiplicity

Let $k$ be an algebraically closed field, and let $A$ be a Buchsbaum homogeneous integral domain over $k$. Say that $A$ has minimal multiplicity of degree $q$ when its minimal multiplicity degree is $q$.

Nonexistence conjecture. There is no Buchsbaum homogeneous integral domain over $k$ with minimal multiplicity of degree $q \ge 2$.

The preceding proposition establishes this assertion for minimal multiplicity of degree at most $2$ under the stated hypotheses, while the existence of examples for degree $q \ge 2$ remains unresolved in the source.