A finite word w is called closed if it has length at most 1 or it contains a proper factor that occurs both as a prefix and as a suffix but does not have internal occurrences. An infinite word u is called closed-rich if the infimum of all possible ratios between the number of closed factors within any factor w of u and square of the length of w exists and is positive. We define this infimum as the closed-rich constant \(C_u\) of the infinite closed-rich word u. Puzynina and Parshina (2024) proved that infinite closed-rich words exist. In this paper, we estimate possible values of \(C_u\) for an infinite closed-rich word u, and apply these results to estimate the supremum \(C_{sup}\) of the closed-rich constants of infinite closed-rich words. We show that \(0.0952 < C_{sup} \le 0.165964\) , where the lower bound comes from the Fibonacci word.