Large-denominator conjecture for continued fractions avoiding squares

Let $v$ be a non-square integer with $v\geq10^{13}$. Consider rational numbers with continued fraction expansions

$$[0;4a_1,4a_2,\ldots,4a_n,a_{n+1},1,2],$$

where the $a_i$ are positive integers. Large-denominator continued-fraction conjecture. For every such $v$, there exists a coprime positive integer $u<v$ such that

$$\frac{u}{v}=[0;4a_1,4a_2,\ldots,4a_n,a_{n+1},1,2].$$

The conjecture is motivated by computation and concerns a family whose denominators have no square values despite having no corresponding local obstruction.