The conjecture on the least exponent for double almost squares

Let $g(\theta)$ be the least exponent such that, for some $c_1,c_2>0$, every interval of the form

$$[x-c_1x^{g(\theta)},x+c_1x^{g(\theta)}]$$

contains an integer $n=a_1b_1=a_2b_2$, where

$$a_1<a_2\leq b_2<b_1$$

and all four integers lie in

$$[x^{1/2}-c_2x^\theta,x^{1/2}+c_2x^\theta]$$

and $0\leq\theta<1/2$; the constants $c_1$ and $c_2$ may depend on $\theta$. The double almost-square exponent conjecture. For $1/4\leq\theta<1/2$,

$$g(\theta)=1-2\theta.$$

The conjecture is motivated by the proved lower bound $g(\theta)\geq1-2\theta$ and the upper bound $g(\theta)\leq1-\theta$ for $1/4\leq\theta\leq1/3$; no resolution is supplied here.