Coprime-factor congruence conjecture in short intervals

Let $\theta$ satisfy $1/2<\theta\leq 1$. Let $q$ be a positive integer and let $c$ be an integer with $(c,q)=1$. Coprime-factor congruence conjecture in short intervals. There is a constant $C_\theta$ such that, for every $N\geq q^\theta$, the congruence

$$xy\equiv c\pmod q$$

has solutions satisfying

$$C_\theta N\leq x,y\leq 2C_\theta N,$$

and $(x,y)=1$. This is presented as a variation of the preceding congruence conjecture; the supplied text gives no resolution.