Chains of large gaps between primes

Let $p_n$ denote the $n$-th prime, and for any $k\ge 1$ and sufficiently large $X$, define the quantity $$G_k(X):=\underset{p_{n+k}\le X}{\max }\min (p_{n+1}-p_n,\dots ,p_{n+k}-p_{n+k-1}),$$ which measures the occurrence of chains of $k$ consecutive large gaps of primes. Recently, with Green and Konyagin, the authors showed that $$G_1(X)\gg \frac{\log X\log \log X\log \log \log \log X}{\log \log \log X}$$ for sufficiently large $X$. In this note, we combine the arguments in that paper with the Maier matrix method to show that $$G_k(X)\gg \frac{1}{k^2}\frac{\log X\log \log X\log \log \log \log X}{\log \log \log X}$$ for any fixed $k$ and sufficiently large $X$. The implied constant is effective and independent of $k$.