Continued fraction conjecture for $-\pi/4$

A continued fraction has partial numerators $a_n$ and partial denominators $b_n$, with the displayed initial terms determining the pattern. Continued fraction conjecture for $-\pi/4$. The constant $-\pi/4$ admits the continued fraction representation

$$-\frac{\pi}{4}=\cfrac{1}{-1+\cfrac{1^2}{-3+\cfrac{2^2}{-5+\cfrac{3^2}{-7+\dots}}}}$$

where

$$a_n=(n-1)^2,\qquad b_n=-(2n-1),\qquad n\in\mathbb{N}.$$

This is presented as a non-canonical continued-fraction identity in the context of symbolic induction and the search for alternative representations of transcendental constants; the supplied material gives no evidence that it has been proved or disproved.