Conjecture on convergence of the Flint-Hills series at prescribed irrationality measure

Let $u,v>0$ satisfy $1+\frac{u}{v}>2$. Let $\mu(\alpha)$ denote the irrationality measure of an irrational real number $\alpha$, let $\mathcal{P}$ be a sine-like function with period $\alpha$, and let $\mathcal{S}_{u,v}$ and $\mathcal{A}_{u,v}$ be the series defined in the paper.

Convergence conjecture. There is an irrational $\alpha$ with

$$\mu(\alpha)=1+\frac{u}{v},$$

such that, for every sine-like function $\mathcal{P}$ with period $\alpha$, the series $\mathcal{S}_{u,v}$ converges, and consequently $\mathcal{A}_{u,v}$ converges as well.

The preceding theorem constructs, for every $u,v>0$, an irrational $\alpha$ of the same prescribed irrationality measure for which both series diverge. The author therefore conjectures that convergence can also occur when $1+\frac{u}{v}>2$; the source provides no resolution of this claim.