A question about maximal non-valuation subrings

In this paper we pursue and deep the study of ring extensions \({R \subset S}\) such that R is a maximal non-valuation subring of S Ben Nasr and Jarboui in Houston J Math, 2009 (in press)]. It is proved in Ben Nasr and Jarboui Houston J Math, 2009 (in press), Theorem 3.2] that if R is integrally closed with finite Krull dimension, then R is a maximal non-valuation subring of qf (R) iff R is not local and |R, qf (R)]| = dim(R) + 3. This result encourages us to pose the following question: Let n be a nonzero positive integer greater than 2 and let R be a finite-dimensional domain such that |R, qf (R)]| = dim(R) + n, does there exists an overring S of R such that R is a maximal non-valuation subring of S? This paper deals mostly with this question. We solve this question in case R is integrally closed.