The exponent pair conjecture and conjectural formula for beta

Let $(k,\ell)$ be an exponent pair when it satisfies the exponent-pair estimate, and let $\beta(\alpha)$ denote the associated exponent defined by the growth of the model exponential sums. The exponent-pair region is the triangle

$$0\leq k\leq \frac12,\qquad \frac12\leq \ell\leq 1-k.$$

Exponent pair conjecture. Every point in this triangle is an exponent pair. In particular, $(0,1/2)$ is an exponent pair. Moreover,

$$\beta(\alpha)= \begin{cases} \alpha/2,&0\leq\alpha\leq 1,\\ \alpha-1,&\alpha>1. \end{cases}$$

This conjecture describes the expected optimal exponent-pair region and the corresponding sharp growth exponent for the associated exponential sums; the source gives no resolution status.