Density of D5 unit shapes on the hypercycle arc

Determine whether the set of unit shapes of rank-2 Minkowski unit lattices arising from D5-extensions of Q with a unique real embedding is dense along the specific hypercycle arc in the modular surface GL_2(Z)\backslash H defined by the circle (x + 1/2)^2 + (y - 1/(2\sqrt{3}))^2 = (2/\sqrt{3})^2, between the points (1 + i\sqrt{3})/2 and -1/2 + i\,5/(2\sqrt{3}).

Background

In the quintic case p = 5, the authors prove that the unit shapes of D5-extensions with a unique real embedding lie on a single hypercycle arc in the modular surface GL_2(Z)\backslash H. This curve is given explicitly as an arc of the circle (x + 1/2)^2 + (y - 1/(2\sqrt{3}))^2 = (2/\sqrt{3})^2, with endpoints (1 + i\sqrt{3})/2 and -1/2 + i\,5/(2\sqrt{3).

Computational experiments presented in the paper indicate that the unit shapes populate this arc extensively, suggesting a density phenomenon. However, establishing such density rigorously remains unresolved. This question is analogous in spirit to density problems studied for unit shapes in other families (e.g., totally real cubic fields), but here it is posed for D5-extensions constrained to a specific hypercycle.