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