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We show that any quasi-Gorenstein deformation of a $3$-dimensionalquasi-Gorenstein Buchsbaum local ring with $I$-invariant $1$ admits a maximalCohen-Macaulay module, provided it is a quotient of a Gorenstein ring. Such aclass of rings includes two instances of unique factorization domainsconstructed by Marcel-Schenzel and by Imtiaz-Schenzel, respectively. Apart fromthis result, motivated by the small Cohen-Macaulay conjecture in primecharacteristic, we examine a question about when the Frobenius pushforward$F^e_*(M)$ of an $R$-module $M$ comprises a maximal Cohen-Macaulay directsummand in both local and graded cases.

Remarks on the Small Cohen-Macaulay conjecture and new instances of maximal Cohen-Macaulay modules