Existence and evaluation of limiting densities for factorial-square representations

Determine whether the limits of D_k(X)/X exist for k=4, 5, and 6, and characterize their values as Euler products or convergent series if they exist.

Background

The paper defines D_k(X) as the number of integers n≤X for which the least number of distinct factorials whose product is a square and whose largest argument is n equals k. It proves the order of growth D_4(X) asymp D_5(X) asymp D_6(X) asymp X, but does not establish asymptotic densities for these sets.

The unresolved issue is whether each normalized counting function D_k(X)/X converges, and, if so, whether the resulting constants admit explicit descriptions through Euler products or convergent series.