Irrational strict local maxima of the fractional-part autocorrelation

Definitions and mathematical statement

Write $\{x\}=x-\lfloor x\rfloor$ for the fractional part of a real number. For $\lambda>0$ define \[ A(\lambda)=\int_0^\infty\{t\}\{\lambda t\}\,\frac{dt}{t^2}. \] This improper integral converges.

Does there exist an irrational $\lambda_0>0$ and a number $\varepsilon>0$ such that \[ \lambda>0,\quad 0<|\lambda-\lambda_0|<\varepsilon \quad\Longrightarrow\quad A(\lambda)<A(\lambda_0)? \] The comparison includes every nearby positive real argument.

Short English statement

Can this fractional-part autocorrelation function have a strict local maximum at a positive irrational number?

Sources

M. Balazard, contribution to Dirichlet Series and Function Theory in Polydiscs, Oberwolfach Report 6/2014, p. 350, Question 2, https://doi.org/10.4171/owr/2014/06. The question is recalled there from L. Báez-Duarte, M. Balazard, B. Landreau and E. Saias, Étude de l'autocorrélation multiplicative de la fonction "partie fractionnaire", Ramanujan Journal 9 (2005), 215--240. This is an independently written formulation of their question.