Let $\rho^\ast(x)$ denote the maximal length of an admissible sequence in $[1,x]$, i.e. of a sequence which does not cover all the residue classes modulo any $n\geq 2$. Hensley and Richards (1974) showed that $\rho^\ast(x)>\pi(x)$ for $x=20000$, hence under the prime tuples conjecture we also have $\pi(x+y)>\pi(x)+\pi(y)$ for the same $x$ and some $y$.

Clark and Jarvis (2001) found that already $x=4916$ satisfies $\rho^\ast(x)>\pi(x)$. Are their any smaller values of $x$ available, has anyone worked on this question since? Clark and Jarvis (2001) also showed that $\rho^\ast(x)\leq \pi(x)$ for $2\leq x\leq 1426$.

Added. As Wojowu remarked, $x=3159$ is also known to satisfy $\rho^\ast(x)>\pi(x)$.