Logarithmic evaluation conjecture for a family of generalized continued fractions

For an integer $k\geq 1$, consider the generalized continued fraction

$$\mathcal{C}\left[\frac{n^2}{k^2(1-4n^2)}\right].$$

Logarithmic evaluation conjecture. For all $k\geq 1$,

$$\mathcal{C}\left[\frac{n^2}{k^2(1-4n^2)}\right]=\frac{2/k}{\operatorname{ln}(k+1)-\operatorname{ln}(k-1)}.$$

These identities give explicit limits for a family of continued fractions that had previously been known to converge but whose limits were unknown. The supplied source reports that the formulas were verified numerically and describes them as conjectures; no proof or later resolution is given here.