On a problem related to “second” best approximations to a real number

For a given irrational number $\alpha$ one can define an irrationality measure function $\psi_{\alpha}^{[2]}(t) = \min\limits_{\substack{(q, p)\colon q, p \in\mathbb{Z}, 1\leqslant q\leqslant t, \\ (p, q) \neq (p_n, q_n) ~\forall n\in\mathbb{Z_{+}}}} |q\alpha -p|$, related to the second-best approximations $\alpha$. In 2017 Moshchevitin studied corresponding Diophantine constant $\mathfrak{k}(\alpha) \liminf\limits_{t\to\infty} t \cdot\psi_{\alpha}^{[2]}(t)$ and spectrum $\mathbb{L}_2 \{ \lambda ~| ~\exists\alpha\in\mathbb{R\setminus Q} \colon \mathfrak{k}(\alpha) \}$. particular, he calculated two largest elements of $\mathbb{L}_2$. present paper we calculate value for third element