Twin Ramanujan-prime ratio bounds

Let $\pi_2(x)$ count twin-prime pairs up to $x$, and let $\pi_{2,1}(x)$ and $\pi_{2,2}(x)$ count, respectively, twin-prime pairs with one and with two Ramanujan primes, as defined in the paper.

Twin Ramanujan-prime ratio conjecture. For all $x\ge10^5$,

$$\frac{\pi_{2,1}(x)}{\pi_2(x)}<\frac45,\qquad \frac{\pi_{2,2}(x)}{\pi_2(x)}>\frac25,\qquad \frac{\pi_{2,2}(x)}{\pi_{2,1}(x)}>\frac{2/5}{4/5}=\frac12.$$

These inequalities sharpen the earlier prediction that more than one quarter of twin primes are twin Ramanujan primes. They are supported by the tabulated computations, while the supplied status evidence leaves them open.