The conjectural density of twin primes is $\frac {c\cdot n}{(\log n)^2}$ at a $c>0$.

Consider integers of form $p,p+1=2^tq,p+2=r$ where $p,q,r$ are primes and $t\geq1$ holds.

1. Is there any reason to believe there are infinite of them at a given $t\geq1$? Is there a conjectural density for such triples at a given $t$?

2. Is there any reason to believe there are infinite of them with $t$ not fixed? Is there a conjectural density for such triples with $t$ not fixed?