Short-factor congruence conjecture

Let $\epsilon>0$ be any small positive real number, let $q$ be a positive integer, and let $c$ be an integer with $(c,q)=1$. Short-factor congruence conjecture. The congruence

$$xy\equiv c\pmod q$$

has solutions with $1\leq x,y\ll_{\epsilon}q^{1/2+\epsilon}$. This conjecture is presented as related to the rational-sum approximation problem, but no resolution is given in the supplied text.