Sinai's positive entropy conjecture for the standard map

For each parameter $\lambda$, define the standard torus diffeomorphism

$$f_\lambda(x,y)=(2x-y+\lambda\sin 2\pi x,x).$$

Sinai's conjecture. For every parameter $\lambda\ne 0$, the metric entropy of $f_\lambda$ is positive. This is presented as another famous positive-entropy conjecture; the source does not report a resolution.