Density of squarefree integers requiring exactly four factorials

Determine whether the squarefree integers n with F(n)=4 have density zero, thereby establishing whether D_4(X)/X tends to 1-6/π².

Background

The paper proves that nonsquarefree integers generally admit a four-factorial square representation and obtains the lower bound liminf D_4(X)/X ≥ 1-6/π². The remaining contribution to the density of D_4 comes from squarefree integers with F(n)=4.

Erdős and Graham suggested that the squarefree integers with F(n)=4 might have density zero. If that conjectural density statement were true, then the limiting density of D_4 would equal 1-6/π². The paper’s two-block argument requires a large prime factor of n and therefore does not resolve the issue.