A Diophantine inequality involving different powers of primes of the form $[n^c]$

Let $[\, x\,]$ denote the integer part of a real number $x$. Assume that $λ_1,λ_2,λ_3$ are nonzero real numbers, not all of the same sign, that $λ_1/λ_2$ is irrational, and that $η$ is real. Let $\frac{219}{220}<γ<1$ and $θ>0$. We establish that, there exist infinitely many triples of primes $p_1,\, p_2,\, p_3$ satisfying the inequality \begin{equation*} |λ_1p_1 + λ_2p_2 + λ_3p^4_3+η|<\big(\max \{p_1, p_2, p^4_3\}\big)^{\frac{219-220γ}{208}+θ} \end{equation*} and such that $p_i=[n_i^{1/γ}]$, $i=1,\,2,\,3$.