On a Diophantine inequality with prime numbers of a special type

We consider the Diophantine inequality $|p_1^c+p_2^c+p_3^c-N|<(\log N)^{-E}$, where $1<c<15/14$, $N$ is a sufficiently large real number and $E>0$ is an arbitrarily large constant. We prove that the above inequality has a solution in primes $p_1$, $p_2$, $p_3$ such that each of the numbers $p_1+2$, $p_2+2$ and $p_3+2$ has at most $[369/(180-168c)]$ prime factors, counted with multiplicity.