I am reading the proof of Theorem 1(a) in the paper that proposed the CLIME method for estimating precision matrix. I am puzzled by an inequality on Page 605 three lines above formula (29). I isolate the specific question as follows for your convenience.

Let $\mathrm{X} = (X_1,...,X_p)^T$ be a $p$-variate random vector. $\{\mathrm{X}_1,...,\mathrm{X}_n\}$ is an iid random sample from the distribution of $\mathrm{X}$. WOLG, assume $\mathbb{E}=0$. Define $Y_{kij} = X_{ki}X_{kj} - \mathbb{E}(X_{ki}X_{kj})$. Suppose the following exponential-type tail condition holds: there exists some $0<\eta<1/4$ and a bounded constant $K$ such that $\log p/n\leq\eta$ and $$\mathbb{E} e^{\lambda X_i^2} \leq K<\infty, \quad \mbox{for all } |\lambda|\leq\eta, \mbox{ for all } i\in \{1,...,p\}.$$ Let $t=\eta\sqrt{\log p /n} $. Prove that $$ n t^2 \mathbb{E}\left(Y_{kij}^2 e^{t|Y_{kij}|} \right) \leq \eta^{-1}K^2\log p$$ holds.