Suppose that $B_{j,n}$ and $C_{j,n}$ are two triangular arrays of iid Bernoulli variates, with each $B_{j,n}$ independent of each $C_{j,n}$ and
$$ \mathbb{P}[B_{j,n}=1]=p,~\mathbb{P}[B_{j,n}=0]=1-p,~\mathbb{P}[C_{j,n}=1]=q,~\mathbb{P}[C_{j,n}=0]=1-q. $$
I define, recursively, the sequence
$$ \begin{cases} K_{0,n}&=&0\\ K_{j,n}&=& B_{j,n}+B_{j,n}\,K_{j-1,n} \end{cases} $$
The integer number $K_{j,n}$ represents the total number of consecutive $B_{q,n}=1$ with $q\leq j$. Similarly I define
$$ \begin{cases} Q_{0,n}&=&0\\ Q_{j,n}&=& C_{j,n}+C_{j,n}\,Q_{j-1,n} \end{cases} $$
I want to prove that, for given $j$ and $k$,
$$ 0\leq \text{cov}(K_{j,n}\,\mathbb{I}\left\{K_{j,n}\leq Q_{j,n}\right\},K_{j+k,n}\,\mathbb{I}\left\{K_{j+k,n}\leq Q_{j+k,n}\right\})\leq \text{cov}(K_{j,n},K_{j+k,n}\,)\quad(1) $$
where $\mathbb{I}\left\{\cdot\right\}$ is the indicator function. Below I report some preliminary computations, however they do not seem to help me very much in proving (1) (which I verified via simulations). Is there maybe a simple argument using some property of the indicator function?
Some attemps First note that
$$ K_{j,n} = \begin{cases} 0 & \text{ with probability }1-p\\ 1 & \text{ with probability }(1-p)\,p\\ 2 & \text{ with probability }(1-p)\,p^2\\ \vdots & \vdots \\ j-1 & \text{ with probability }(1-p)\,p^{j-1}\\ j & \text{ with probability }p^j \end{cases}. $$ and similarly for $Q_{j,n}$. Hence
$$ \mathbb{E}[K_{j,n}] = 0\,\cdot (1-p)+1\,(1-p)\,p+2\,(1-p)\,p^2+...+(j-1)\,(1-p)\,p^{j-1}+j\,p^{j}=\frac{p \left(1-p^j\right)}{1-p} $$ and similarly $$ \mathbb{E}[K_{j,n}^2]=\frac{p\, \left(1+p-(2 j (1-p)+1+p) p^j\right)}{(1-p)^2} $$
whence
\begin{array} \mathbb{E}[K_{j+k,n}\,K_{j,n}] &=& \mathbb{E}[\left(B_{j+k,n}+B_{j+k,n}\,B_{j+k-1,n}+...+B_{j+k,n}\,B_{j+k-1,n}\,\cdots B_{j+1,n}K_{j,n}\right)\,K_{j,n}] \\ &=& \mathbb{E}[K_{j,n}]\,p\,\frac{1-p^{k-1}}{1-p}+p^{k}\,\mathbb{E}[K_{j,n}^2] \end{array}
so that
$$ \text{cov}(K_{j,n},K_{j+k,n}) = p\,\frac{\left(1-p^{2 j}-2 j \,(1-p) p^{j-1}\right)\,p^{k+1}}{(1-p)^2} $$
Now consider that
\begin{array} \mathbb{E}[K_{j,n}\,\mathbb{I}\left\{K_{j,n}\leq Q_{j,n}\right\}]& = &\mathbb{E}[\mathbb{E}[K_{j,n}\,\mathbb{I}\left\{K_{j,n}\leq Q_{j,n}\right\}]|K_{j,n}] \\ &=&\mathbb{E}[K_{j,n}\,\mathbb{E}[\mathbb{I}\left\{K_{j,n}\leq Q_{j,n}\right\}]|K_{j,n}]\\ &=&\frac{p \left(p^j q^j (j (q-1) (p q-1)+(p-1) q)-p q+q\right)}{(p q-1)^2} \end{array}