Continuity and strict monotonicity of Nyman–Beurling distances

Definitions and mathematical statement

Work in the Hilbert space \[ H=L^2((0,\infty),t^{-2}\,dt),\qquad \|f\|_H^2=\int_0^\infty |f(t)|^2\,\frac{dt}{t^2}. \] Let $\chi$ be the indicator of $[1,\infty)$ and put $e_\alpha(t)=\{t/\alpha\}$, where braces denote fractional part. For real $\lambda\geq1$, define \[ D(\lambda)=\inf\{\|\chi-v\|_H: v\in\operatorname{span}\{e_\alpha:1\leq\alpha\leq\lambda\}\}. \] The span uses finite linear combinations over $\mathbb R$; allowing complex coefficients or taking its closure gives the same distance.

Is $D$ continuous on $[1,\infty)$, including right-continuity at $1$? Is it strictly decreasing there, so that $D(\lambda_1)>D(\lambda_2)$ whenever $1\leq\lambda_1<\lambda_2$?

Short English statement

As the permitted range of fractional-part dilations expands, is the distance to the indicator function continuous and strictly decreasing?

Sources

M. Balazard's second and third questions in the problem session of Dirichlet Series and Function Theory in Polydiscs, Oberwolfach Report 6/2014, p. 390, section 8; reporter O. F. Brevig, https://doi.org/10.4171/owr/2014/06. This is an independently written formulation of the two questions.