  Let $\tilde K_t$ denote the $3$-uniform linear clique of order $t$. Given an
even integer $t \geq 4$, let $M$ denote the asymmetric maximal density of
$\tilde K_t$ and $\tilde K_{t/2}$. We prove that there exists a constant $C>0$
such that, if $(H_n)_{n \in \mathbb{N}}$ is any sequence of dense $3$-uniform
hypergraphs and $p := p(n) \geq Cn^{-1/M}$, then $$ \lim_{n \to \infty} \Pr[H_n
\cup R_n \to (\tilde K_t)_2] =1 $$ holds, whenever $R_n \sim \mathbb{H}(n,p)$,
where the latter denotes the binomial $3$-uniform random hypergraph
distribution. We conjecture that this result uncovers the threshold for the
property in question for $t \geq 6$.
  The key tools of our proof are a new variant of the Strong Hypergraph
Regularity Lemma along with a new Tuple Lemma accompanying it. Our variant
incorporates both the strong and the weak hypergraph regularity lemmas into a
single lemma; allowing for graph-like applications of the strong lemma to be
carried out in $3$-uniform hypergraphs.
