Does this prime-gaps pattern occur infinitely often?

Question

Let $p_n$ be the $n$-th prime.

For each integer $k \ge 0$, do there exist an infinite number of $k+3$ consecutive primes $(p_n, p_{n+1}, \ldots, p_{n+2+k})$ so that

• (1) The gap between the 1st and 2nd, and between the 2nd and last, are equal: $p_{n+1}-p_n = p_{n+2+k}-p_{n+1}$.

• (2) There are $k$ primes between the 2nd and last, i.e., between $p_{n+1}$ and $p_{n+2+k}$.

For $k=0$, the answer is Yes by the recent breakthroughs on prime gaps. Here are some examples:
     
One could whimsically imagine "skipping" a flat stone on the primes, where the first bounce covers the gap between the 1st and 2nd primes, followed by $k+1$ smaller bounces that together cover the same gap before sinking on the last prime.

Answer 1

This would follow from the $k$-tuple conjecture in the following way.

Choose an admissible tuple $d_1, \ldots, d_{k+2}$, such that $d_2-d_1=d_{k+2}-d_2$. If $n\in[d_1, d_{k+2}]$ is an integer, such that $\{d_1, \ldots, d_{k+2}\}\cup\{n\}$ is admissible, pick a prime number $p$ such that for all $i$ we have that $n-d_i$ is not divisible by $p$. Pick integers $e_i>d_{k+2}$, such that $\{d_1,\ldots, d_{k+2}, e_1, \ldots, e_\ell\}$ is admissible, and the set $\{d_1,\ldots, d_{k+2}, e_1, \ldots, e_\ell\}$ covers all residue classes modulo $p$ with the exception of $n\pmod{p}$. Repeat until you arrive at an admissible set $D$, such that $D\cap[d_1, \ldots, d_{k+2}]=\{d_1, \ldots, d_{k+2}\}$, and for all $n\in[d_1, d_{k+2}]\setminus\{d_1, \ldots, d_{k+2}\}$ we have that $D\cup\{n\}$ is not admissible. Apply the $k$-tuple conjecture to the set $D$. If $x+d$ is prime for all $d\in D$, and $x$ is sufficiently large, then $x+d_1, \ldots, x+d_{k+2}$ are consecutive prime numbers.