Density conjecture for the real parts of complex dimensions of nonlattice strings

Let $\mathcal{L}$ be a nonlattice string, with $D$ its Minkowski dimension and $\sigma_l$ the lower bound for the real parts of its complex dimensions.

Density conjecture. The real parts of the complex dimensions of $\mathcal{L}$ form a set that is dense in the connected interval $[\sigma_l,D]$.

This conjecture strengthens the preceding result that the real parts of the complex dimensions of every nonlattice string have no isolated points, asserting density throughout the entire interval from $\sigma_l$ to $D$.